aleph cardinality paradox

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Paradoxes in Transfinite Arithmetic and the PerceivedInconsistency of ZF Set Theory

B. M. Smith

Innovative Nuclear Space Power and Propulsion InstituteUniversity of Florida

PO Box 116502, Gainesville, FL 32611-6502, USAEmail: [email protected]

April 2003

Abstract

Perhaps no other area of contemporary mathematics invites quite as many doomsayersas does set theory and related foundational mathematical theories dealing with the conceptof the infinite. This paper looks at recent articles posted on the World Wide Web (WWW)that claim that Zermelo-Fraenkel set theory is inconsistent. Although other purportedproofs have been posted on the Web, this paper considers a particular class of proof thatcenters around the operation of exponentiation of infinite numbers. The common resultcited is that 20 = c 0, which seems to contradict another theorem of transfinite settheory that |2| || = || = 0. The arguments that claim such results yield internalcontradictions within ZF set theory are examined and shown to be false and based upon acomplete misunderstanding of set theoretic principles. A brief outline of possible reasonswhy many people perceive set theory to be inconsistent is given. The paper is written fora student readership for mainly pedagogical purposes. No original theorems are derived.Derivations are informal yet tight enough in logical structure to enable students moderatelyfamiliar with set theory to check the derivations in more rigorous formal language at theirleisure.

In a series of sweeping studies on the foundations of mathematics the logician Kurt Godelsucceeded in proving that any axiomatization of the self-evident (intuitive) laws of arithmeticin a formal system, with sufficient power to prove theorems, is doomed to be either inconsistent

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The empty set is a member of all sets. Other sets are defined or proven to be sets inductively.The symbol is the set of natural numbers {0, 1, 2, . . .}. That is actually a set (and not justa collection of numbers) is an axiom of ZF theory (the Axiom of Infinity), so it is a postulate inthe same sense that is simply assumed to be a set. It is not trivial that an infinite collection of

sets (each natural number can be proven to be a set) is itself a set. However, the reasonablenessof the postulated sethood of was established by Dedekind long before ZFT was thought of,indeed, Dedekind succeeded in defining a totally new class of mathematical objects (sets) thatcould support a one-to-one mapping into a proper subset of the original set, and proved thatno finite set could support such a mapping. Thus the abstract existence of infinite sets wasknown before ZF set theory was formulated. The Axiom of Infinity was therefore not at allunreasonable or unwarranted. To some readers it may seem strange that anyone should evenquestion the right of mathematicians to call the collection of all natural numbers a set. Whileone can sympathize with the simple intuition that all the natural numbers can be conceived ofas a unity, it would be nicer not to have to make this an axiom, one would rather like to prove

it from simpler intuitions. ZFT takes the position that infinite sets can be motivated abstractlywithout any reference to finite objects, and thus that the Axiom of Infinity is simply therebyheld to be non-controversial.

Yet the question can be legitimately asked, can an inconsistency be derived from thisaxiom? A formal inconsistency would be, at a minimum, a pair of theorems, one showing thatsome statement S is a theorem, and one showing that not S is a theorem. This is also calledan internal contradiction. Note that such a contradiction is entirely distinct from the type ofcontradiction generated in proofs using the method of reductio ad absurdem (RAA). In proofsemploying RAA one never assumes the truth of statement S, even though teachers often usesloppy English and in a proof will write now lets assume S. . . and proceed to generate theinconsistency, this is bad form, because in RAA what one really does is prove a more compact

statement like (S (T)) where T is some previously established theorem of the theory, andT is the negation of this theorem. In other words, the author of the proof never actuallyassumes that statement S is true of the current axiomatized theory, but rather, the authoris in effect imagining a logically possible universe in which all the axioms of the theory holdin addition to the statement S as a new axiom and then the proof proceeds to work out thepurely logical consequences of this counterfactual world. If a logical inconsistency ensues thenwe can know that in any consistent mathematical universe the statement S must therefore befalse (where consistency is relative only to the prior postulates and axioms of the theory).

Any such inconsistency in an axiomatic scheme endowed with a suitable formal language. Inset theory the language of predicate calculus1 is used in formal proofs. If a logically inconsistentset of statements could be derived as theorems it would be catastrophic because it would renderthe axiom scheme useless, inasmuch as any statement whatsoever could then be deduced as a

1Devised by Peano this language consists of all well-formed (grammatical) statements that can be made fromsome simple atomic sentences (axioms or assumed true postulates and simple objects of the universe of discourse)using the seven basic symbols (negation), (or), (and), (implies), (if and only if), (for all), (thereexists, or for some). A comma or colon is sometimes added for such that, and parentheses are used to formcompound well-formed sentences.


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valid theorem by including in a derivation the conjunction (S (S)) or S and (not S). Inthe so-called 0 paradox the inconsistency can be pithily summarized in two statements, one isthat 20 = c > 0, and the other is

00 = 0 < c. The aim of the first part of this paper is

to show that there is no such inconsistency derivable in ZF set theory, at least not in any form

given by the claimants [1, 2].The axioms and basic theorems of ZF set theory are assumed as established as valid deduc-

tions, but the consistency of the ZF axioms is not assumed. So this paper does not assume thatZF theory is consistent. Indeed, the purpose is to examine this as a distinct possibility. A proofof the inconsistency of ZF set theory would indeed be highly welcome, because it would forcemathematicians to look for perhaps better foundations for modern axiomatic mathematics.

2 Description of the 0 Paradox

This section presents the main purported proofs of inconsistencies in ZFT. In the following

should be strictly interpreted as the ordinal number, the least ordinal greater than all thenatural numbers. But without cause of error one can also interpret loosely as the set = Nthe set of all natural numbers.

Conjecture 1. 1. By Cantors diagonal argument theorem 20 0 is a theorem of ZFT.

2. It is also a theorem of ZFT that || = || = 0, so in particular || 20.

3. A third theorem of ZFT is that 00 = c = 20.

The conjecture is therefore that ZFT (|||| 0)(|||| 0), yielding an inconsistencyin ZFT.

This conjecture is actually false, so a proof is not given. Instead the following attemptedproof is taken almost verbatim from the references as an illustration of how thinking abouttransfinite set theory can go astray.

Proof. (Attempted.) The idea is that for exponentiation of cardinals a and b one finds |ab| =|a||b|. Then by results 2 and 3 we have |||| = 0 c. So ZFT (||

|| 0).But by result 3 we also have 00 0, or in other terms, ||

|| 0. So ZFT (|||| 0).Therefore ZFT (|||| 0) (|||| 0), as required.

This type of argument is given for example in [1] and [2]. The error in this type of argument isobvious to anyone with some formal background in transfinite set theory. To provide sufficient

background for understanding the error the following sections present a cursory summary oftransfinite arithmetic. Section 4 on page 16 will explain the simple error for any reader whocannot see it immediately.


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3 Cardinal and Ordinal Numbers

Dispensing with a lot of technical formal development, the reader is now invited on a whirl-windtour of the arithmetic of finite and infinite sets. For background the reader is referred to either of

the references [3, 4, 5]these references are ordered in decreasing level of technicalityreference[5] is the easiest to read though still fairly technical in parts.

3.1 Ordinal Arithmetic

Having established the existence of infinite sets in the abstract, and then having found a goodcandidate for an infinite set, namely , the natural question is to ask whether or not moreinfinite sets can be constructed. The first way of creating new infinite sets is to just continuethe process that was used to build up the entire set of natural numbers, namely by continuingto make new sets by appending the entire collection of previous sets to itself. This is the processof formation of ordinal numbers.

This is a very important notion. One might think that adding a small number like 1 to aninfinity like cannot make a larger number. But that is a mistaken interpretation of whatordinal counting is doing. In ordinal counting one merely asks what order or arrangement ofabstract sets can be built up, so adding 1 to is only shorthand for saying that there is asuccessor set to the set called +. Mathematicians label this set + 1 in analogy with theway successors of natural numbers are named. Thus strictly in ZF set theory 3={0, 1, 2}, butfor shorthand it is often written as 2+1. If this idea of ordering numbers is kept in mind, andadding +1 to get the next ordinal number is understood as merely a convenient shorthandexpression, not an actual arithmetical operation, then the idea of ordinal numbers should bemuch clearer. The confusing thing is that most people still refer to this as ordinal addition, so

in a sense we are stuck with this unfortunate name for the process. Its not all that bad thoughbecause it works out to be the same as true addition for natural numbers. This is an importantpoint because, as we will soon see, it is not the same for infinite numbers, indeed 1 + = + 1is a fact that follows from the strict definition of this ordinal addition.

Abstractly speaking, all ordinals correspond to ordered sets, such that an ordinal numbera would be specified by giving an example of a set S such that if one could count S in thecorrect order then one would count up to the ordinal a. Then the ordinal corresponding to someset A is found by ignoring the actual appearance of the individual members of A and insteadconcentrating on their order or arrangement. Then a + b is obtained by first counting to a andthen counting b steps further. The ordinal a b is obtained by counting to a and then doingthis again b many times. The exponential ab is obtained by (i) counting a, then counting again

another a steps repeating this a many times, then (ii) repeating (i) b many times.However, these three types of ordinal operation (addition, multiplication and exponentiation

of ordinals) are defined with the order of doing things strictly in mind, so that ba is conceptuallydistinct from ab and likewise for a+b and b+a. Counting to a first then counting b steps furtheris clearly different to counting to b first and then counting a steps further. The end result isthe same for natural numbers (and for all finite numbers), but conceptually one is obtaining the


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same result but in two different ways (you can run the 800m by going twice around an Olympictrack either clockwise or anticlockwise). Some concrete examples are given in the followingpages.

Thus one starts with 0 = , then append this to itself to make a new set called 1, so

1 = {, {}}, then append this to itself to make the set called 2, so 2 = {, {}, {, {}}},this process is continued indefinitely to create all the naturals numbers, which by the Axiom ofInfinity is itself a set, = {0, 1, 2, 3, . . .}. So the next ordinal number after is simply writtenas,

+ 1 = {0, 1, 2, 3, . . .} {} = {0, 1, 2, 3, . . . , } (1)

Three things are notable here. First, + 1 is just a name, it does not mean add +1 to , andas one can clearly see by the righthand side of this definition the concept of this named set iscompletely sensible. But it is very closely analogous to adding 1, by the interpretation of thisset as the successor to . The second thing is that instead of the boldface symbol here the

plain symbol is used. This is the convention chosen in this paper to refer to the set of naturalnumbers as an ordinal number, rather than specifically as a set. This is purely a matter of styleand taste for the present, but there are good reasons why the set and the ordinal number formed from this set should be conceptually distinguished, even though it makes little differenceif one uses the same symbol for both. The third notable thing is that the last comma in Eq.( 1)is put in to indicate that all the natural numbers go before in this set, whereas appearing inthis set is a completely new addition, it is a single entity and a subset of the new entity +1.So now that this new entity + 1 is established the process can continue and we get,

+ 2 = {0, 1, 2, 3, . . . , , + 1}

+ 3 = {0, 1, 2, 3, . . . , , + 1, + 2}...

+ = 2.

(2)

The two ideas expressed here are (i) that when you have an ordinal number a you can find a nextordinal a + 1, and (ii) when there is a sequence of increasing ordinals a + 1, a + 2, a + 3, . . . thenthere is a last ordinal that is greater than all these that could be expressed as lim( a+n) = a+.Strictly speaking the +1, +2, and +3 here do not have to be necessarily the naturalnumbers, they could be any other sequence of ordinals. So continuing the list begun in Eq.(2)we could have, 2 + 1 , 2 + 2 , . . . 2 + = 3, where after the dots we indicate reaching

yet another limit ordinal. Continuing on we get: 3+1, 3 + 2, . . . , 3 + = 4, . . . , 5,. . . , 6, . . . , = 2. In this last stretch a lot of intermediate ordinals have been skippedover. To speed things up the next infinite sequence of limit ordinals can be conceptually jumpedthrough to yield another limit ordinal as follows, first we get the sequence 2, 3, . . . . Afterthis limit we can skip through the sequences +1, +2, . . . + = 2.

Next jump through the sequences 2, 3, . . . = 2

, another limit ordinal. Now jumpthrough these as follows,

2

, 2+1, . . .

2+, . . . 2+2, . . .

2+2 = 22, . . .

3

, . . .

.


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At each limit here one imagines speeding up the progression to the rate of the previous limit,so that sequences of lower limit ordinals are entirely skipped over. Eventually we can jumpthrough another sequence as follows,

, +1 . . . ,

2, . . . ,

2, . . . ,

, . . . .

. .

= , + 1, . . .

where the last limit ordinal here is a really big one with many exponentiations, which is called tetrated to the power of , and after the next ordinal + 1 the . . . indicates that westill havent stopped. Indeed, this way of progressing through the ordinal numbers never reachesan exhaustion point. Unlike each limit ordinal there is no actual limit of all the ordinals. Thiscan be seen by considering the collection Ord of all ordinal numbers. Is this collection itself anordinal number? If it is then Ord + 1 is an ordinal number not in the collection of all ordinalnumbers, a contradiction. So by reductio ad absurdem there is no set of all ordinal numbers.

The hierarchy of ordinal numbers is fascinating, but the main concern of this paper is themore mundane seeming topic of how to add and multiply these ordinal numbers, since this iswhere the initial inclings of paradox emerge. The first sign of something funny is that ordinaladdition turns out to be non-commmutative for the infinite ordinals, even though ordinal addi-tion is commutative for finite numbers. This is easy to see in the simplest case. Consider thefollowing results,

1 + = {1} {0, 1, 2, . . .} =

+ 1 = {0, 1, 2, . . .} {} = {0, 1, 2, . . . , } = 1 +

Both of these results can be obtained rigorously from set theory, but the short answer to why1 + = + 1 is evident in the way ordinal numbers were defined by the two rules (i) and (ii)above. We obtain the next ordinal always by adding 1 (technically, by forming a new setfrom the previous ordinal by joining that set with the previous ordinal itself), but in the otherorder of operation the set {1} is joined with an ordinal, but since {1} is already a member ofthe set of all successor ordinals greater than 1 we gain nothing by taking the union of manynatural numbers with {1}. On the other hand, / and is a limit ordinal = lim(1 + n),so adding 1 to gives a next ordinal. Its all a matter of definition. One does not have to likethe outcome of this process for obtaining higher and higher ordinals, but one cant escape themathematical fact that the procedure is valid and gives some sort of rigorous type of transfinitearithmetica conceptually distinct mathematical structure to finite arithmetic that a priorineed not be assumed to follow the same rules as finite arithmetic.

Intuitively though it doesnt really seem all that strange. By loosening the formalism a little,just imagine 1 + as conceptually adding many 1s to 1 where 1 could stand for anything,apples, circles, planets, steinlagers, or whatever. So in, 1 + =1+(1+1+1+. . . ), clearly this isno different than simply adding together many 1s, so the result is just 1 = . For thereverse order operation just imagine + 1 as adding 1 to an already reached limit of many1s, so + 1=(1+1+1+. . . ) +1, and this is conceptually, if not intuitively, different from theformer because in this case one has to imagine having first summed up to , so there is no


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next natural number that can be obtained by continuing to add 1s, thus a new ordinal mustbe obtained, + 1 = . Again, the difficulty some students have with this results from themisinterpretation of ordinal addition as literally adding up numbers, which is wrong, what weredoing is examining arrangements of numbers in ordered sets. Were not even trying to perform

addition in the traditional sense. If you like to think of it as addition then thats fine, it is ageneralized sort of addition. Here are some more general assorted results.

1. + = 2 = 2 = , so two lots of is quite different to many 2s.

2. + 2 = {, + 1, + 2, . . . , 2, . . . , 3, . . . , n . . .} = lim( n) = 2. So thereis no point in having apples if you are just going to add 2 more of them!

3. 2 + 2 = 2 2 = 2 2 = 2. So two lots of 2 gives a lot more steinlagers than 2

many two-packs of steinies!

4.

2

+

3

=

3

=

3

+

2

=

3

+

2

.5. 6 + 3 3 + 6 = 6 2.

6. 10 + 1010 + + 5 = + 5.

Further, as an exercise the reader might like to try adding these three ordinals in all sixpossible orders, = , = + 5, = 2. There are five different results that should beobtained: 2, 2 + + 5, 2 + 2 + 5, 2 + , and 2 + 2.

These results follow from the definitions of the natural numbers on the one hand and thedefinition of ordinal addition on the other. Now what about ordinal multiplication?

Ordinal multiplication is also non-commutative. But this is parasitic on the non-commutativity

of ordinal addition, so nothing essentially new is discovered. Multiplication has already beenemployed in many of the results above. However, grouping is important. Here are some moreassorted results, displayed conceptually with the set theoretic derivations implicit.

1. 2 = lots of 2s = 2+2+2+. . . = 1+1+1+1+1+1+. . . =.

2. 2 = 2 lots ofs = (1+1+1+...) +(1+1+1+...) = 2 > .

3. = + + + . . . = lim( n) = 2.

4. ( + 3) = 2.

5. ( + 3) = 2 + 3 = ( + 3) .

6. ( + 3) ( + 2) = 2 + ( + 3) + + 3) = 2 + 2 + 3.

7. ( + 1) 3 = + 1 + + 1 + + 1 = 3 + 1.

8. 3 ( + 1) = (3+3+3+. . . )+3 = + 3.


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For all these results one gets the right answers with the knowledge that is a transfiniteordinal. If were just a symbol then none of the above results could be derived unambiguously.Normally one would just assume that any undefined symbols can be treated as finite numbers, sosymbolic variables then revert to obeying the good old rules of commutative arithmetic. When

the use of transfinite ordinals is clear from context then addition and multiplication no longercan be dealt with as trivially as before. If the symbols are undefined then one also wouldnt knowwhether to treat them as complex numbers, Grassmann variables, multivectors, octonions, orwhatever. The distinction between finite and infinite sets is a categorical one, entirely analogousto the distinction between real numbers and complex numbers.

Subtraction and division of ordinals are defined using addition and multiplication, but aswith the natural numbers, subtraction is only defined for when > . In that case it ispossible to prove that there is a unique ordinal number that satisfies the equation = + .For example, ( + 3) 2 = + 3, but (3 + ) 2 = , uniquely. And ( + 1) = 1 uniquely,and 2 = because + = 2.

Similarly, as long as > 0 one can prove that for any other ordinal there exists uniqueordinal numbers and such that = + , where 0 < . When = 0 it is said that is a left divisor of , and is a right divisor of .

Cardinal Exponentiation

Finally ordinal exponentiation should be considered (logarithms of ordinals will be left for theexperts). First note that a result used above was that = 2. The ordinal 2 here canalso be thought of as a limit ordinal, namely the first ordinal a such that + a = a. To seethis, note that = + + + . . ., so its easy to see that putting an extra + in frontof this makes no difference to the arrangement. Now in a similar way = . . ., so

another way of thinking of

is as the first ordinal a such that a = a, since putting in front of a = . . . will make no difference to the order. The general definition ofordinal exponentiation is a bit like this, it starts inductively by (i) defining 0 0 = f0(0) = 0 anda0 = fa(0) = 1 for any ordinal a > 0. Then (ii) to get at a

b first consider all the successorordinals in b, call them o+, then fa(o

+) = fa(o) a, and finally (iii) for every limit ordinal inb, define fa() = {fa() : < }, then rules (i), (ii), and (iii) define ab as follows,

ab = fa(+). (3)

The notation {f(x) : R(x, y)} used here means the union of all elements of sets defined bythe function f(x), over all sets x that have a relation R(x, y) with the set y. In rule (iii) the

relation is ordinally less than, and the union is of all elements of the function fa() on x.Thus ordinal exponentiation is defined recursively from the bottom up. To evaluate ordinalexponentials you go backwards from the top down, seeing which rules apply. To see that thisworks out as you would expect from the previous rules for ordinal multiplication, consider thefollowing examples.


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1. 3 = f(2) , by rule (ii), since 3+ = 4 is not a limit ordinal

= f(1) , rule (ii) again

= f(0) = 1 , by rule (i)= .

2. 3 = f3(+), by rule (iii), since is a limit ordinal

= {f3(n) : n < }, still by rule (iii)

= {f3(0), f3(1), f3(2), . . .}, by rule (ii) successively applied

= {1, 3, 9, 27, . . .} = .

3. = f(+), by rule (iii), since is a limit ordinal

= {f(n) : n < }, still by rule (iii)

= {f(0), f(1), f(2), . . .}, by rule (ii) successively applied

= {1, , 2, 3, . . .}, which itself is a limit ordinal as previously

deduced heuristically.

practise for

1. 5+1 = f5( + 1) = f5() 5 = {h5(0), f5(1), f5(2), . . .} 5 = {1, 5, 25, . . .} = 5.

2. (+1)3 = f+1(3) = f+1(2)(+1) = f+1(1)(+1)(+1) = 1((+1)(+1))(+1)= (2 + + 1) ( + 1) = (2 + + 1) + (2 + + 1) 1 = 3 + (2 + + 1) =3 + 2 + + 1.

3. ( +2)+1 = f+2( +1) = f+2()( +2) = {1, +2, ( +2)2, . . .}( +2) = ( +2)

= +1 + 2.

4. (2 + ) = . . . = {1, (2 + ), (2 + )2, . . .} = (2) = 2 = .

5. 33+5 = f3( 3) 35 = f3( + + ) 243 = f3()f3()f3() 243 = ()()() 234 =3 243.

The last few of these examples were done rather heuristically.

A good exercise for the student is to prove that the following laws hold (order still important)(1) ab+c = ab ac, and (2) (ab)c = abc. Other provable rules like these, such as n = for allnatural numbers n 2, help to ease the calculation of ordinal exponentials.

The main thing is that for finite ordinals the results of ordinal addition and multiplication arecommutative and obey all the familiar rules of arithmetic for natural numbers, and for transfiniteordinals addition, multiplication, subtraction, division and exponentiation are non-commutativeyet entirely well-defined, no contradictions arise.


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3.2 Cardinal Arithmetic

Now, the finite ordinals are just (isomorphic to) all the natural numbers. But a set like ={0, 1, 3, 5, 7, } is not a natural number. One would still like to perform arithmetic with such

finite sets. In order to define a generalized addition for finite sets that are not natural numbersa new notion is required. This will be the concept of the cardinality of a set. It will enable ageneralized concept of size to be defined for all sets, and cardinal arithmetic will turn out to bedefinable on all finite sets, not just the natural numbers.

In order to do this the rank of a set is defined recursively by a function r() as follows. Leta be any fixed ordinal number. Let y = f(x) be a formula. The for any ordinal number b andsuccessor ordinal b+, define the function ra() as follows

ra(0)df= 0.

ra

(b+)df= f(r

a(b)), whenever b+ < a.

ra(b)df= {ra(x) : x < b}, whenever b is a limit ordinal in a.

Then define the rank of a set X to be ifX r(+)+(+) \ r+() = r(

+) \ r(). Here thesymbol \ denotes set subtraction or relative complementation. If such an ordinal exists thenX is said to have rank and has rank .

In more down to earth language, the rank of a set can be viewed as a function from sets toordinal numbers. The rank of a set is then the least ordinal number greater than the rank ofany member of the set. A few important results about rank can be stated (without proof here,but see for example [4, chapter 6] for proofs).

1. Every ordinal number has itself as its rank.

2. If < then r r.

3. If a set has a rank then it is well-founded.

4. The Axiom of Regularity is equivalent to the statement that every set has rank.

5. The Axiom of Regularity guarantees that every ordered set corresponds to a unique car-dinal number.

Some examples: The empty set has rank 0 (since it has no members and 0 is the least ordinal

number). The set 1={{}} has rank 1 (since its only member {}, has rank 0). The set {{{}}},which is not a natural number, has rank 2 because 2 is the least ordinal greater than the rankof this sets member element 1. The set {{}, {{}}, . . .} has rank .

Now the reason why this concept of rank is important is because it turns out to be very usefulfor defining the cardinality of sets without resorting to a controversial axiom called the Axiom ofChoice. Instead a much weaker axiom can be adopted: the Axiom of Regularity, which assertssimply that every set is well-founded. A set A is well-founded if every nonempty subset B


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of A contains an element b for which b B = . Unfortunately this axiom restricts set theorysomewhat because collections of objects like all the ordinal numbers cannot be consideredas sets, because they would violate the Axiom of Regularity. But such collections are alreadyknown to be paradoxical, so not much power is lost by adopting the axiom of regularity. The

natural numbers are all well-founded, and so is . These sets are sufficient for development ofa huge load of powerful mathematics and applications.

So far, the result of most interest for transfinite arithmetic is item 5 on this list. Yet theconcept of a cardinal number has not yet been defined. This is now remedied.

First, let X = Y or X Y indicate that there is a bijective map from the set X into theset Y.

For any set A, there is a set called the cardinal of A written |A| defined to be the collectionof all sets X of lowest rank for which X = A. This is the technical definition of the cardinalof a set without reference to the Axiom of Choice. A less formal definition would be that thecardinal of a set A is the least ordinal a such that a = A, where a is viewed as the set {o : o < a}

of ordinals o that are less than a.In practise the result that is most often used is the establishment of a bijective correspondencebetween sets X Y that therefore proves that |X| = |Y|. Alternatively one may write X = Y.All these three statements can be taken as equivalent.

To compare cardinals a special type of order symbol is used to unambiguously distinguish theordering from the totally distinct ordering relations , and used for ordinal numbers.So for any two cardinals 1 and 2 on writes 1 2 whenever there are sets A1 and A2 with|A1| = 1, |A2| = 2 such that there is an injective mapping of A1 into A2. The converse alsoturns out to be true. Furthermore, if 1 = 2 then one can write 1 2. It is obvious that inthese cases respectively an alternative cardinally greater than is to write 2 1 iff there isan injection A1 A2, and when 1 = 2 one could write 2 1.

Having defined what a cardinal is it is then possible to talk about arithmetic of cardinals,specifically cardinal addition, multiplication and exponentiation. Since it is primarily cardinalexponentiation that is of interest later on, the following discussion of cardinal arithmetic will bebrief and even more informal than the previous discussion of ordinal arithmetic.

Cardinal addition is fairly intuitive. If1 and 2 are two cardinals, then to find the cardinal1 + 2 one simply finds two sets K1 and K2 such that |K1| = 1 and |K2| = 2. Then define1 + 2 = |K1 + K2|.

The conceptual difference between ordinal and cardinal addition is this. To find the cardinal2+3 find a set X with 2 elements and find a set Y with 3 elements, then find the smallest ordinala such that the set {o : o < a} can be put in bijective correspondence with X Y. Whereas tofind the ordinal 2+3 by ordinal addition first count to two then count a further three numbers.

To define cardinal multiplication, suppose again that 1 and 2 are two cardinals, the we de-fine 1 2 to be the cardinal of the Cartesian product 12. Note that all cardinals are first andforemost sets, so their cartesian product is always defined. For reference, the Cartesian productof two sets X and Y is the set of pairs {a, b : a X, and b Y}. For example the Cartesianproduct of the ordinals 2 and 3 is the set 23={0, 1}{0, 1, 2}={0, 0, 0, 1, 0, 2, 1, 0, 1, 2}.But note that the Cartesian product of natural numbers will not always be a natural number.


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This is why cardinal arithmetic is useful, it can be employed on any sets, not just ordinals,because every set will have a corresponding unique cardinal (or cardinality) according to theAxiom of Regularity. (Note of course that all cardinals are limit ordinals by definition.)

Now the raw results of both cardinal multiplication and cardinal addition turn out to be

the same as ordinal addition for finite numbers. However, for transfinite numbers cardinal andordinal addition are completely different and so it is advisable to use a different symbol forthe cardinal form of arithmetical operations. Accordingly in the rest of this paper when twocardinals are added this will be indicated by the bold symbolic operator +++++++++, and likewise whentwo cardinals are multiplied this will be indicated by the symbolic operator , thus the followingsymbolic notation will be used

+++++++++ , tells us implicitly to add these as cardinal number.

+ , tells us implicitly to use ordinal addition.

, tells us implicitly to use cardinal multiplication.

, tells us implicitly to use ordinal multiplication.

(4)

The key difference for transfinite sets is that when one of or or both are infinite then +++++++++ = max(, ). This is a very mundane result and it makes cardinal multiplication oftransfinite numbers a fairly easy job. It is quite different to ordinal multiplication. For example,ordinally + = 2 = , but cardinally one gets ||+++++++++ || = || = 0. The law for transfinitecardinal addition follows because all infinite sets support a noninjective surjection, so the smallercardinal can always be mapped to the larger cardinal with enough elements left-over in the largerset to still map bijectively this subset of the larger cardinal to itself.

A similar result holds for transfinite cardinal multiplication. The Cartesian product can be

imagined as the square or rectangular array with the first row having all the elements of one ofthe cardinals , and the first column having all of the elements of the second cardinal . TheCartesian product defining is then the array formed by pairing the ith first row entry withthe jth first column entry, forming an infinite array whenever one or both of or are transfinitesets. A bijection between the largest of these cardinals and the whole Cartesian product arrayis obtained by first noting that = , because any transfinite cardinal can be bijected toits own Cartesian self-product by simply filling in the Cartesian product array with ordinalschosen from (every cardinal is a limit ordinal, and hence is an ordered set). This is done byfilling in the array by mapping the ordinals in in order starting at the 0, 0 top left cornerentry of the Cartesian product array and then successively traversing the next column to theright going down to the diagonal entry and then traversing that entrys row to the left until

hitting the leftmost entry (as in the proof of the denumerability of the rationals from the mapN N Q). So a bijection is established , thus = as required.

Now because = the result = max(, ) must follow as a corollary. To seewhy, just note that the Cartesian product array for can just as easily be traversed bysuccessively mapping elements from the larger of these cardinals (say for arguments sake) inthe same way as in the proof of the result for . The asymmetry of the array in thiscase is of absolutely no concern.


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Cardinal Exponentiation

Unlike cardinal addition and multiplication, cardinal exponentiation poses a challenge. Expo-nentiation of ordinals was fairly-straight forward recall. In the case of ordinals the extension

of multiplication to exponentiation was natural and similar to the way finite exponentiation isdefined in terms of multiplication. For cardinals one has to generalize from the rather differentidea of multiplication as the formation of a Cartesian product of sets. How is this accomplished?

Intuitively could be thought of as adding up many objects, then adding another objects to get twice , and repeating this in total many times to get objects. In otherwords, we are multiplying many products of s. In terms of proper cardinal multiplicationwed translate this as = . . . ( many s here). This will be an array of dimension. To simplify the conception of this infinite dimensional array the preferred thing to do is torecognize that such an array can be considered as isomorphic to all sequences of length that canbe formed from any elements of. Call such sequences Seq(), thus these are all the sequencesthat are possible by choosing elements of over and over again until one has a sequence of

terms. This can be viewed as a set isomorphic to . Thus for example, all members of Seq3(2)can be written as,{0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1}. Thereare eight members of this class of sequence, just as one expects for a representation of |23|.

Using this definition the following results can be proven (in the following, , , and areall assumed to be cardinals).

1. For all cardinals , |1| = . Also, |0| = |1|, and |0||0| = |1| also. Also |0| = |0|, becausein this case the domain for the -sequence is empty.

2. Since the set 1 = {0} only has one member then |1| = |1|, because there is only oneunique -sequence that can be formed, i.e. |Seq(1)| = |1|.

3. +++++++++ = .

4. ( ) = .

5. () = +++++++++.

6. If1 3 and 2 4, then 21

43 .

7. If 2 3 then it may be possible that a 1 can be found such that 21 =

31 . For

example, for all natural numbers n , |||n| = || = 0.

8. There are examples known of cardinals 1, 2, 3, 4, such that 1 3 & 2 4, suchthat 21 = 43 .

9. There can be cardinals found for which 1 3 such that 1 = 3 . For example, a well

known result is |2||! | = |R|| ! | = |R|. (|R| is the cardinality of the real numbers, alsodenoted c.)


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10. For all natural numbers n such that 2 n then |n||! | = |2||! |. Here can beconsidered as the ordinal or the set of natural numbers.

11. For every set X, |(X)| = |2||X|. So in particular |()| = |2||! | = c, by result 9.

12. If0 is the first transfinite cardinal. Formally one gets 0 = ||, which is sensible since is the first transfinite ordinal.

13. A special result is that |2| = || = 0, where within the || cardinal operator one performsordinal exponentiation.

14. It can also be proven that, c 0, which is Cantors diagonal argument theorem, in aneven fuller account this means, c = |2|0 = |2|

|omega| 0.

So the solution for defining cardinal exponentiation is quite natural. The problem is thatnone of the more interesting results yield cardinals of known cardinal order. It is known that the

set of all cardinals is well-orderable, so any two distinct cardinals and are related by either or . But when asking simple questions about how the exponentials of cardinals arerelated it is not always the case that an unambiguous answer can be given. For a start we havea well-ordered set of finite cardinals, the natural numbers. Then all the transfinite cardinals aredenoted (in increasing order) 0, 1, 2, . . . , +1, . . . and so on. The ordinals can be used toindex all these alephs which are the ordered cardinals.

The problem lies in interpreting the exponentials of cardinals as cardinals. Navely onewould expect the exponentiation of cardinals results in another cardinal. But one of the verysimplest nontrivial transfinite cardinal exponentiations, namely |()| = |2||! | = c is of unknowncardinality. This is the so-called Continuum Problem. Cantor originally conjectured that c =1, which is one version of the Continuum Hypothesis (CH). But this later turned out to be

unprovable within ZFC. In 1940 Godel showed that there is a model of ZFC in which |()| = 1,this meant that CH is consistent with ZFC, so that one can never prove that c = 0. But thatonly means that Cantor was not provably wrong within the axioms of ZFC. In 1963 Paul Cohenshowed that the negation of the CH is also consistent with ZFC, he did this by showing thatthere is a model of ZFC in which () = 2, and there is also a model in which () = 3, andindeed there are a whole lot of models that together mean that () can be almost any alephwithin reason. Together Godels and Cohens results say that ZFC is not powerful enough todetermine uniquely the cardinality of () = c.

Another more philosophical version of a continuum hypothesis, that perhaps far more aptlybears the same name, is that, c = |R| is the cardinality of the continuous line. It is very hard

to know whether the real numbers can fill in a line completely. It is known that the reals forma metrically complete set, but no one knows if this can be used to infer that the infinity of thereals is so big that this many point sets can actually fill in a continuous line. In fact, due tothe result that c c = c this means that c (or equivalently |2||| or |()|) many points mustalso be numerous enough to fill in any other continua like the 2D plane, or a 3D volume andso on. The flip side of this is the question of whether a point (a single number) is actually ofabsolutely zero width or extent, or is it some infinitesimal extent, smaller than any real number,


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but not absolutely zero? Its even possible that for most applications it does not matter whatthe cardinality of points in a continuum is, it may even be consistent to have absolutely infinitelymany point in any given continua. As long as a chosen set of points like the real numbers isdense and complete on the line, or R R plane, etc., then probably most useful mathematics

can be done by assuming the points fill in the continua, but without having to explicitly assumeso. In this light, the question really is, what abstract property of a continuum is so vital thatdenseness and completeness of point set topologies on the continuum will not suffice to impartthis property? This is the question for which no one has so far provided a good intuitive answer.

The theory of hyperreal numbers and nonstandard analysis may have answers to these ques-tions. However, this subject is not really related to the more mundane Continuum Hypothesis(CH) of cardinals, which is a more mathematically well-defined question. So the interestingtopic of the cardinality of points forming a line is usually not explored by standard set theory.

The mathematical CH, that 20?

= 1, however is actively pursued by set theorists, andextensions of ZFC are sought for which the CH may be provably true of false based upon some

extra simple intuitions. So far none have been found that are universally agreed upon. Sothe CH remains an open question in mathematics. Some people are even of the opinion thatmathematics need not decide on only one consistent universe, so that both CH and CH maysimply define alternative mathematical frameworks of equal validity. Again, this is a questionthat is not universally agreed upon.

Transfinite Arithmetic

If ordinal and cardinal arithmetic was not enough, it turns out that infinite summands andinfinite products can also be defined, but not just boring old sums like

i=0 and products

i=0,

but rather real transfinite cumulative sums and products, in other words, sums or products of a

sequence where terms in the sequence can be functionally dependent on any transfinite ordinal,such as 2 + 3 or whatever. So if is any transfinite ordinal we can get a transfinite sum

or product, written as say


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be patently clear to the attentive reader why the attempted proof of Conjecture 1 is invalid.The short answer is that ordinal and cardinal arithmetic have been thoroughly confused in theattempted proof. In fact results of ZFT have actually been used inconsistently in the purportedproofs. To some extent this is an inexcusable error, but allowance for possible excuses for these

errors are discussed later in Section 5.So the short answer why the paradox is no a paradox, and certainly not an inconsistency is

really in the first line of the attempted proof where it is state that |ab| = |a||b| for any cardinalsa and B. This is then used invalidly to infer that || = ||||. This is obviously invalid becauseon the left ordinal exponentiation is implicitly adopted, whereas on the right it is clearly cardinalexponentiation that is implied.

It is also obvious that countless paradoxes and inconsistencies would result if one wereto similarly equate ordinal arithmetical operations with cardinal arithmetical operations. Toattempt to say that there is only one valid generalization of finite arithmetic for transfinitenumbers is a rank amateur sort of attempt to consider transfinite sets on an equal footing with

the completely distinct mathematical structures of finite sets. There is no a priori philosophicalbasis for such attempts. This is likely to be the case for any generalization of arithmetic totransfinite sets given the characteristic distinctness between the finite and the infinite.

Equally obvious is the seeming fact that this explanation is not satisfying to some whocontinue to see a paradox in the totally confused mix-up between ordinal and cardinal expo-nentiation. The conceptual difference between these operations is simply not factored in to anyphilosophical thinking by such critics of set theory. Some possible reasons why they might insistthat a paradox and inconsistency remains are discussed in Section 5 below.

Notice that on page 16 above a number of alternative statements of the CH were given.

c = 1,

|2||| = 1,

|2|0 = 1,

20 = 1,

|()| = 1,

(5)

are all equivalent statements of the CH. The 3rd and 4th statements use the fact that 2 is a setas well as a cardinal number. So it should be unambiguous to write 20 as implying the operationof cardinal exponentiation rather than ordinal exponentiation. The reason being that 0 is acardinal, so when used in the exponent one naturally assumes that cardinal exponentiation isimplied. However, writing 2 is not the same as writing |2||| because in the former case the

ordinal exponentiation operation is implied, while in the latter case an unambiguous cardinalexponentiation operation is implied. And indeed the results of these respective operations arecompletely different, thu8s 2 = , thus |2| = || = 0 which does not equal |2||| 1, or ifthe CH is adopted as an axiom, then in ZFT+CH one would have |2||| = 1.

So it is entirely consistent in ZFT to have |2||| = 1 and |2| = 1. In these and otherso-called aleph paradoxes one is never in the position of deriving statements S and not(S)


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both as theorems. To suppose that |2| = 1 is equivalent to not(|2||| = 1) is a grossamateur mistake and oversight of mathematical language and logic.

This in essence is the source of the first paradox. If one inconsistently (falsely in fact)assumes that ordinal and cardinal exponentiation should yield the same results then the paradox

is ensured. This is of course totally misguided and a violence against logic and reason tosuppose that cardinal and ordinal exponentiation should be the same. If it were the case thenone wouldnt need exponentiation of alephs and s to arrive at a paradox, simple transfinitemultiplication and addition would suffice.

This completes the refutation of the conjectures given in Section 2.

4.1 Why are These Results Thought of as Paradoxes?

OK, so all should be clear now that no paradoxes have been proven in ZFT. There is no incon-sistency in the formal system of ZF set theory, at least not as claimed by the cited references.

So what was the basic problem or fundamental motivation that may have driven people awry?The real question is, why would anyone who understands set theory be so contrarily disposedto the theory to reach at such desperate straws to vainly hope for some imagined death blowcontradiction, or so acerbic in regard for what was originally a beautifully rebellious branch ofmathematicsa quintessence of novelty and brilliant expanding of the human mindto actuallywant so badly to find an inconsistency where none exists?2

There seem to be two identifiable objections to ZF set theory that are alluded to in theaborted attempts to prove inconsistency.

1. The results just dont look right and should be rejected for reasons of aesthetic taste orphilosophical presumption.

2. There is some deeper mathematical intuition telling us that even transfinite numbersshould not behave this way, indicating a serious error in the very conception of the infinite(starting with the Axiom of Infinity perhaps).

Philosophical objections of type 1 are highly dubious and amount to straight out prejudiceand an ad hoc attitude to mathematics whereby one accepts a bunch of primitive axioms onlyas long as they imply consequences that ones delicate philosophical intuitions can swallow,at which point one simply chooses to abandon the axioms because they are obviously wrongwith no justification other than the apparent ugliness of the theorems to the eye of the beholder.Nevertheless, some of the saner philosophical objections along these lines will be discussed laterin Section 5 below. To put things in a starker light, the only aspects of a mathematical frameworkthat should be questioned are its axioms, not its valid theorems. If the theorems look strangeand counterintuitive then this may be reason to call the axioms into question, but unless a true

2It should be stated again here that this paper does not claim that ZF set theory is consistent, and thebrilliance of the theory is not taken in any way as a reason why mathematicians should not be looking for betterfoundations. For even if the ZF axioms are consistent there may still be a more satisfying and pleasing foundationfor mathematics that just needs inventing.


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inconsistency is found one cannot exactly object to the axioms as long as they are in themselvesreasonable and intuitive.

Such is the case with ZFT, most mathematicians find the axioms fairly intuitive and self-evident. In this case the supposedly unpalatable theorems about transfinite sets are either true

inconsistencies or they have to be accepted and absorbed into the consciousness for meditationin case they are indeed revealing profound aspects of the formalization. No one says that ZFTis consistent, but until actually proven inconsistent there is no reason to believe that ZFT isleading the mathematical community down a proverbial garden path.

Type 2 seems to be the more valid objection. One could ask for instance, suppose ZF settheory is consistent, that doesnt mean it has to be the only formalization of human mathematicalintuition, so maybe there is another consistent formal system that captures human intuitionsbetter and yields a less counterintuitive form of transfinite arithmetic and algebra?. Section 5also makes an attempt to touch upon this type of objection to transfinite set theory.

Yet in this case, if transfinite arithmetic turned out to obey the same rules as finite arithmetic,

and if ordinal and cardinal arithmetic gave equivalent isomorphic results then this would be inall likelihood a better reason to prefer ZFT to the fictional alternative. Why should one preferthe more counterintuitive theory? The short answer is that if ordinal and cardinal operations ontransfinite sets were no different to the same operations on finite sets then there would in factbe little difference at all between finite and infinite. This would be far more counter-intuitivethan having the purported paradoxes of ZFT. Its not that one prefers paradoxes, but ratherthat in the case of finite versus infinite one fully expects the patterns and rules of the numbersto be totally different. It is simply outrageous to expect that a theory of the infinite can bebased in a theory of the finite. Yet those who seek a form of set theory that is grounded in theidea that patterns and rules that hold for finite sets as limits are approached should thereforeextend to infinite sets are essentially asking for just such a counter-intuitive set theory.

So again, what is the deeper intuition that tells some people that transfinite arithmeticshould obey similar rules that hold for finite sets in the limit that the infinite is approached?Well, there does seem to be some merit in supposing that if relations between finite numbershold right up until the infinite is actually reached then the same rules shouldnt just magicallychange when the infinite is actually attained. This is the exact position adopted in [2]. Thereare two comments that could be offered in response to such intuitions. First, one would preferto base such an intuition in simpler axioms, and not have to take this sort of behaviour as anaxiom. Secondly, as was the case for the Type 1 objection, there does not really seem to be anyphilosophical supremacy for holding such a view in the first place.

Consider that in actually imagining abstractly attaining the infinite one is in some sensegoing beyond all finite realms of number patterns. One could ask, Why should there not besome radical change in behaviour of the generalized numbers that are reached after ? Inthis light is a definite dividing line between two classes of ordinals, and 0 is a clear divisionbetween two classes of cardinals. Why should they not define a sudden catastrophic changein behaviour? Such catastrophes are littered throughout science, why not in mathematicstoo? Indeed, much of the current research in chaos theory, complexity and catastrophe theoryemploys mathematical models that demonstrate such jumps in characteristic behaviour and


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sudden magical changes in pattern that one sees described by the ZFT version of change ingoing from the finite to the infinite.

5 Philosophical Objections to Set TheoryIt might be argued, by some, that the results of following through the logic of the axioms of ZFset theory yields such strange and counterintuitive results that we (the collective community ofmathematicians) should just dump set theory and start again. Why cant we have a set theorywhere transfinite addition and subtraction are defined and commutative, and where exponen-tiation is sensible and yields the same results on ordinals and cardinal numbers alike? Theserequests do not seem unreasonable. But are they really all that intuitive and philosophicallyappealing that wed want to hold on to them at all costs?

Well, to answer the critics, it turns out that commutative addition and multiplication oper-ations can be defined on all the ordinals if thats what you like. The operations are called the

Hessenberg natural sum of ordinals and the Hessenberg natural product of ordinals.the idea is simply to use expansions of ordinals in terms of powers of . This is natural inthe same sense that natural numbers are added on an abacus or table in terms of powers of(usually) 10.

Think of what perhaps is one of the most suspicious ideas in transfinite set theory, that onecan remove an object from one set S1 that stands in bijective correspondence with a second setS2, to give a smaller set S1 and yet still be able to find a bijection between the smaller set S1and the unchanged former S2. One can even more counter-intuitively do this when removinginfinitely many objects from S1. The bijection D : E defined by D : k 2k defined for allnatural numbers k where Even is the set of all even natural numbers, is a classic example. We

remove all infinite odd numbers from

and yet end up with a smaller set (the evens) that canstill be mapped bijectively to . Whats going on here?There is a double-barrelled answer to this question. The first is that we are talking about

finite sets all the time up until someone suddenly introduces the infinite set . Then all hellseems to break loose. A smart philosopher might however ask the question in reverse: namely,suppose a newborn babe had grown up having been taught only transfinite number theory, thenall of a sudden the child is exposed to finite number theory and quickly matches up their innateintuitions about finite collections of objects (apples, crayons, lego blocks, etc) with this newconcept of finite numbers. The child is perplexed that all the beautiful theorems of transfinitenumber theory seem to be totally out of kilter and jumbled when translated into finite sets. Ofcourse no human could practically ever experience such a paradigm shock, but essentially every

young mathematician experiences an exactly analogous shock when all the years of instructionon finite number arithmetic is suddenly seemingly overturned by the introduction of transfinitenumbers in their first year university course. The shock is of course illusory, because no resultswhatsoever of finite number theory are abandoned! One is merely being exposed to a completelynew class of sets. This is the proper way to teach the theory. For indeed, transfinite sets areabsolutely distinguised from finite sets. Indeed, Dedekind first established the result in 1888that a set that could be put into a non-surjective injection with itself is fundamentally distinct


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from a finite set (for provably no finite set can be mapped to itself in this way) and that thischaracterizes all infinite sets as different from finite sets.

Moreover Dedekinds characterization of infinite sets in this fashion was completely inde-pendent of any reference to finite sets. So one cannot object the he was in any way placing

artificial man-made restrictions on the mathematization of the infinite, or finitizing the infi-nite as some authors have put it[2]. Quite the contrary in fact. By objecting to ZFT on thebasis of merely counter-intuitive theorems the philosophers who want to abandon ZF set theoryas a foundation for mathematics are really the ones who are finitizing the infinite. They are inessence demanding that patterns and rules that hold for finite sets should also hold for infinitesets. It has been argued in this paper that it is immature to expect all theorems, even in aconsistent mathematical universe of discourse, to conform to intuition. The only results thatthe community of mathematicians could reasonably be asked to get in conformity with intuitionare the results that humans can compute and construct.

Thus there is a fundamental dichotomy by definition, the finite and the infinite. It is in fact

closely analogous to the fundamental dichotomy between the concepts of Zero and One. Howdo you get something from nothing? Start with the empty set. How do we get the idea of 1without assuming that 1 or One is a coherent notion? The answer provided by set theoryis to form the set {} whose only member is the empty set, and call this set 1. The shock ofthis should be roughly the same as the shock in realizing that is fundamentally different fromany finite number, and hence one should not expect in the least that any theorems about finitesets will be at all applicable to infinite sets like .

Note that this method of asking questions in reverse is always a good rubric or methodof philosophy, indeed such methods have been used effectively in physics to resolve apparentparadoxes about time reversibility of the laws of physics and the thermodynamic arrow of time.See for example [9].

In this paper the claim that ZF set theory is free of paradoxes when transfinite sets areintroduced is not being made. The aim is simply to warn students that the counterintuitiveresults of transfinite set theory may in fact turn out to be tremendously profound and farreaching consequences of not just ZF theory but perhaps of any possible formal axiomatizationof human mathematical intuitions! At least of any formalization powerful enough to capturethe very minimal most idea of the infinite.

5.1 Philosophical Issues Related to Cardinality Paradoxes

Similar philosophical statements can be made concerning the Cardinality paradoxes of transfiniteset theory that were made concerning the Golfers paradox. But in the previous exposition of theCardinality parados, there is a more serious issue related to mathematical pedagogy that can beaddressed. This is the issue of lack of rigor in internet postings and other informal expositionsand essays. The intent here is not to criticize the lack of rigor of amateur mathematiciansand essay writers who feel they have something to contribute. But the increasing reliance bystudents upon the easily accessible World Wide Web for readily digestible information threatensto un-educate unwary students. Yet there is a fantastic freedom offered by publishing on


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the Web that shouldnt be withheld, and any serious budding mathematicians will presumablyalways check the facts before being sucked down pits of non-rigorous diatribe. But there mustbe a better golden mean between easy information and factually checked information.

Concerning the Cardinality paradox, the problem clearly consisted of the inability to distin-

guish between different interpretations of the same symbol used in the mathematics literature.Following the style of some authors, this paper has attempted to remove this ambiguity by usingdistinct symbols for the ordering relations. Clearly Web writers need to take some responsibilityfor this as well. I suggest a simple remedy. The following rules or nettiquette are suggested,primarily or authors who are writing about science and technical subjects:

Authors who want to write technical articles and publish them should obey the rule (netti-quette) that all quoted results of technical content and all references to technical literatureshould cite peer reviewed literature.

No assumptions should be made based upon other material published on the World Wide

Web unless that material has been previously published in peer reviewed literature or hasotherwise been similarly checked for veracity by the community fo scholars in some processequivalent to peer review.

An equivalent to peer review could for example be a technical monograph or textbookpublished by any reputable technical publishing house, or any online encylopedia (such asWikipedia) in which the relevant results are quoted and directly referenced.

If these or stronger rules were adhered to, then the present author feels that the mistakesmade in [1] and [6] would not so easily have been made. Mistakes cannot of course be eliminated,but every effort should be made as part of good nettiquette to reduce speculation and flaky logic

in technical literature published on the Web under the guise or claim of serious science. Thebreadth of information and discussion available on the WWW is a treasure and should not beseen as undermining the ability of serious students to search out the facts for themselves andto actively become intellectual police for the Web itself. The fact that mistaken articles arepublished on the WWW concerning fundamental challenges to science and mathematics shouldbe welcomed as an intellectual challenge to set right, but certainly erroneous articles should notencouraged for the mere sake of the challenge!

5.2 A Note on the Sociology of Mathematics

Breifly, the position that most closely sympathizes with the discussions in this paper is the ideaadvanced by Chaitin [7, 8] that mathematics is more like an empirical science than a pure searchfor absolute truths. The old idea that mathematics discovers eternal truths about numbers andother objects is still tenable. But in the light of the 20th century mathematics of Godel, Church,Turing, and Chaitin, it seems clear that the role of human mathematicians is somewhat lessabsolute. The sum of the wisdom that this paper musters on this topic consists in the simplerealization that given a consistent axiom scheme then all the theorems one can derive therefrom


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are indeed eternal absolute truths but only in that circumscribed domain that is traced out by thegiven axioms. It has to be realized that all of mathematics cannot be encapsulated in any finiteset of axioms, and every consistent set of axioms defines a different universe for mathematics.Thus mathematics is blessed with an embarrassingly rich ontology. One can almost choose

what universe to do ones maths in. Humans are however constrained to the exploration of thenecessarily limited realms of mathematics where intuition can be transformed into axiom.

The consequences of these realizations are nevertheless liberating and powerful for the so-ciology of mathematics (how humans interact and function as mathematical explorers). First,it means that young mathematicians need not feel compelled to follow the lore handed downto them by their professors. Secondly, it means that there is truly no end to the depths andwonders that mathematicians have open for exploration. Unlike the physical ocean, the mathe-matical ocean is not only infinite in expanse, but is also infinite in wonder and multi-valuedin its ontology. New theorems are virtually guaranteed to be found that are non-trivial.

6 Conclusions

Hopefully this paper has shown to the beginning student of set theory the importance of for-malizing heuristic arguments, and the role of intuition that goes in concert with formalization.The exploration of completely new foundations for set theory should be encouraged, particularlyin the area of transfinite number theory. More than anything, it is hoped that students willgain insight into the sociological nature of the human mathematical enterprize, and understandthat mathematics is as much about experimentation and hypothesis forming as the branches ofphysical science. Each inequivalent axiomatization of mathematics yields a different abstractuniverse worth exploring. Cantorian, or Zermelo-Frankel theory is just one (possibly consistent)

such universe. Unlike physical science, mathematics is blessed with virtually (and surely literallyas well) infinitely many distinct and wondrous universes in the entire cosmos of mathematics.

References

[1] M. H. Knowles. 2003. The Good Shepherds Paradox: A New Paradox of Infinity in SetTheory. PAIAS, Palo Alto Institute for Advanced Study. http://www.paias.com/paias/-home/Mathematics/. . . /GSP.htm

[2] D. R. Garcia. Counting the Reals: A New Look at Infinities. Private communication, March11-April 7, 2003.

[3] W. S. Hatcher. Foundations of Mathematics, W. B. Saunders, Philadelphia, 1968.

[4] Martin M. Zuckerman. Sets and Transfinite Numbers, Macmillan, New York, 1974.

[5] R. Rucker. Infinity and the Mind: the science and philosophy of the infinite, Bantam Books,New York, 1983.
http://www.paias.com/paias/home/Mathematics/Set%20Theory/GSP/GSP.htmhttp://www.paias.com/paias/home/Mathematics/Set%20Theory/GSP/GSP.htm


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[6] K. Delaney, 2002. A Critique of the Diagonal Method, http://descmath.com/diag/-index.html

[7] G. Chaitin. The Limits of Mathematics: A course on information theory and the limits of

formal reasoning, Springer-Verlag, 1998.

[8] G. Chaitin. The Unknowable, Springer Series in Discrete Mathematics and Computer Sci-ence, Springer-Verlag, 1999.

[9] H. Price. Times Arrow and Archimedes Point: New directions for the physics of time,Oxford University Press, 1997.
http://descmath.com/diag/index.htmlhttp://descmath.com/diag/index.html

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