Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com
When one reads books on mathematics aimed at a popular readership and which eulogize the concept of transfinite numbers, one might get the impression that a clear definition of “cardinal number ” is a simple straightforward matter. For a simple demonstration as to why this is not the case, one might observe the attempt at such a definition at Wikipedia’s formal definition of a Cardinal number where it is admitted that there is no straightforward definition of a cardinal number without relying on abstruse Platonist assumptions such as the axiom of choice and the notion of transfinite numbers.
It should not be surprising that this is the case. For finite sets the definition is quite straightforward and the definition of the cardinal number for any finite set is given by a one-to-one correspondence with a subset of the natural numbers (see Definition D below), whereas for infinite sets, a transfinite cardinal number is given not by the existence of a one-to-one correspondence, but by the absence of a one-to-one correspondence to some other set. Unfortunately for conventional mathematics, this does not actually give a definitive cardinal number for many infinite sets, which means that ad hoc assumptions must be tagged onto this theory of transfinite numbers, see for example Wikipedia - The Continuum Hypothesis.
This rather troubling inherent problem of the transfinite theory of cardinal numbers - together with its inherent contradiction (see below) - is generated by a fundamental axiomatic assumption. That fundamental axiomatic assumption is within the definition:
Definition A: If and only if a one-to-one correspondence exists between two sets then both sets have the same numerical property (termed ‘cardinal number ’) and which satisfies the comparators ‘equals’, ‘less than’ and ‘greater than’. (Footnote: Note that this does not require that for different sets the property must always satisfy the same definition.) (Footnote: Note that there are different versions of this definition, but the fundamental assumption is always included in some way.)
Note B: Note that this definition does not define what the cardinal number for any particular set is, nor how it is calculated.
Note C: Note that by this definition, every set must have some such cardinal number, since for every set there exists a one-to-one correspondence between it and some other set.
The corollary of Definition A, together with Note C, is that if two sets cannot be put in a one-to-one correspondence then they both have cardinal numbers, and those cardinal numbers must be different. Furthermore, it follows from the Diagonal proof that since there is not a one-to-one correspondence between the set of real numbers and the set of natural numbers, then Definition A forces them to have different cardinal numbers. Hence Definition A gives rise to the bizarre contradiction that there can be limitlessly large sets that are smaller than other limitlessly large sets. Note that you will not see conventional mathematics using these terms since they make it so uncomfortably obvious how absurd the contradictory notion is. Instead they use euphemisms like “the set A has a smaller size than the set B ”.
Instead of the fundamental axiomatic Definition A we can simply define a definition without using “If and only if ”. Note that the use of the term “If and only if ” means that one condition (having a one-to-one correspondence) necessarily implies the other (having different cardinal numbers), and vice-versa. But if we simply use “if ” this gives us:
Definition D: If a one-to-one correspondence exists between two sets then both sets have the same numerical property (termed ‘cardinal number ’) and which satisfies the comparators ‘equals’, ‘less than’ and ‘greater than’. (Footnote: As for Definition A, this does not define what the cardinal number for any particular set is, nor how it is calculated; that must be defined separately.)
The corollary of Definition D is that if there is not a one-to-one correspondence between two sets R and N, those two sets can nevertheless have the same cardinal number; this applies if the sets are infinite, and so the cardinal number of both sets is infinite. Furthermore, there can be a one-to-one correspondence between such a set R and a set R2 where both have the same cardinal number, which is infinite (such as where R is the set of real numbers between 0 and 1 and R2 is the set of real numbers between 0 and 2). And there can be a one-to-one correspondence between such a set N and a set Q where both have the same cardinal number, which is infinite (such as the set N of natural numbers and the set Q of rational numbers).
Note that in this case, regardless of whatever cardinal number two sets have in common, in the same way as for Definition A, nothing in Definition D requires that the same definition for ‘cardinal number ’ applies for finite sets and for infinite sets. It only requires that if there is a one-to-one correspondence between two sets, then those two sets have the same numerical property (‘cardinal number’), which need not satisfy the same definition as a common numerical property of two other sets.
This means that we can define that if there exists a one-to-one correspondence between the elements of a set A and a specific proper subset B of the natural numbers (Footnote: For the purposes of this section, all references to natural numbers are to natural numbers excluding zero.) where that subset is the proper subset which includes all natural numbers less than its largest natural number, then we define the cardinal number for the set A to be the largest number in that set B. This defines ‘cardinal number’ for finite sets.
We can also define that if there does not exist a one-to-one correspondence between the elements of a set A and a specific subset B of the natural numbers, then the cardinal number of that set A is infinitely larger than any natural number, and we can apply the term infinite for that cardinal number.
The above gives us only finite cardinal numbers and a single non-finite infinitely large cardinal number, and no contradictions, in contrast to the unfounded Definition A that necessarily implies that different infinite sets can have different cardinal numbers.
See also the Diagonal proof, the secondary argument of the Diagonal proof, and A List with no Diagonal number and Proof of more Real numbers than Natural numbers, and the papers On Considerations of Language in the Diagonal Proof and On the Reality of the Continuum and Russell’s Moment of Candour.
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There is now a paper that deals with a common misconception regarding real numbers, see On the Reality of the Continuum and Russell’s Moment of Candour. Also see the associated paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.
There is now a new page on a contradiction in Lebesgue measure theory.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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Copyright © James R Meyer 2012 - 2018