Introduction - Alex will argue against the absurdity that one can perform inverse operations in transfinite arithmetic. I'm not sure this is possible because subtraction/division is prohibited (and/or meaningless) in transfinite arithmetic.
1. Equinumerous and fewer than - Alex offers helpful definitions of concepts.
a. Equinumerous (ES): if the number of members in set A = the number of members in set B, then A doesn't have fewer members than B.
b. Is Few Than: if the number of members in set A is few than the number of members in set B, then A doesn't equal B.
2. Hume and Euclid - Alex offers further definitions.
c. Hume's Principle (HP): sameness of cardinality = able to be put into one-to-one correspondence.
d. Euclid's Maxim (EM): The whole is greater than the parts.
EM implies that if X is a proper subset of Y, then X has fewer elements than Y. Hence, EM implies a difference in cardinality relative to sets and proper subsets.
3. The Problem - If the sets are infinite, then HP and EM come into conflict. You could have two infinite sets, A and B, that are ES and A has fewer members than B (if B is the set of all natural numbers and A is the set of all even numbers).
4. What is going on? - There's a clash between pre-theoretical intuitions and set-theoretical expressions, argues Alex. You have to reject either (1) HP, (2) EM, or (3) the idea that HP and EM conflict. Keeping all three implies absurdity.
5. Rejecting HP - Alex finds no reason to reject it. Nevertheless, Craig proposes that one can reject it (it could be a mere convention with no a priori proof, but which works excellently for finite collections). To assume it works for infinite collections, argues Craig, is merely an assumption. According to "The Standard of Equality of Numbers" (in Meaning and Method: Essays in Honor of Hilary Putnam) by George Boolos, Hume's principle isn't a 'truth of logic, a definition, an immediate consequence of a definition, analytic, quasi-analytic, or anything of that sort'. In "Finitude and Hume's Principle" (in The Arche Papers on the Mathematics of Abstraction) by Richard G. Heck, Jr affrims that he's an 'agnostic' when it comes to the principle being a 'conceptual truth'; Heck acknowledges that 'Hume's Principle will fail to hold, if it does, only because there are some equinumerous infinite concepts which are assigned to different numbers' and that 'Hume's Principle fails only for infinite concepts.' And according to Kzimierz Trzesicki's "In What Sense is God Infinite?: A Consideration fromt he Historical Perspective" (in God, Time, Infinity ed. Miroslaw Szatkowski), Richard Dedekind, regarding Hume's Principle, 'believed the principle could not be applied to infinite sets.'
6. Rejecting EM - Alex proposes to revise EM (since Alex argues that there might be more to the idea of 'fewer than' than merely being a proper subset):
EM* (applied to infinite sets) - A is fewer than B iff
(1) A is a proper subset of B, and
(2) A and B aren't equinumerous.
In infinite sets, EM*-2 isn't satisfied. Hence, A isn't fewer than B. This implies that if A and B are equinumerous, then, necessarily, A is not fewer than B.
7. Rejecting that HP and EM conflict - In this case, there's no conflict between (1) A being a proper subset of B, and (2) A and B being equinumerous. Hence, there's no conflict between (2) and, (3) A's members being 'fewer than' B's members. To not see this is to be guided, not by set-theoretical mathematics, but pre-theoretical intuitions regarding finite sets. The conflict between HP and EM is a contingent (not a necessary) conflict. Hence, if I subtract a member from a set S of infinite members, the resulting difference will be set S* (with all the members minus the member that was subtracted), where S* has 'fewer' members than S, even though S* and S are equinumerous.
8. Comparison -
- Strategy 1: S* does not have fewer members than S because S* and S are equinumerous (HP).
- Strategy 2: S* does have fewer members than S because S* is a proper subset of S (EM*).
The absurdity of Hilbert's Hotel is motivated by Strategy 2, but not Strategy 1. Hence, if I subtract an infinite number of numbers from S, or if I subtract one number from S, then the resulting difference, S* or S**, will involve two mutually inclusive differences in S because the idea of 'fewer than' and the idea of 'being equinumerous' have been revised in light of set-theoretical mathematics.
9. Conclusion - Either revise EM ('fewer than' isn't in conflict with HP) or appeal to set-theoretical mathematics (over pre-theoretical intuitions with finite sets) that imply the mutual inclusivity of HP and EM (EM's notion of 'fewer than').
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Alex's revision of EM won't help because one of the reasons why Craig argues that Hilbert's Hotel (HH) is absurd doesn't seem to rely on revising EM. Even Wes Morriston in his "Must Metaphysical Time Have a Beginning" (in The Kalam Cosmological Argument, Volume 1 eds. Paul Copan and William Lane Craig) argues:
However, Craig insists that his argument against the actual infinite does not rest on Euclid's Maxim alone. ". . . not all the absurdities stem from infinite set theory's denial of Euclid's axiom: the absurdities illustrated by guests checking out of the hotel stem from the self-contradictory results when the inverse operations of subtraction or division are performed using transfinite numbers. Here the case against an actually infinite collection of things becomes decisive."
And as was said above regarding HP and finite collections, Craig appears to hold the same belief regarding EM:
"Craig acknowledges that in mathematics, 'Euclid's maxim holds only finite magnitudes, not infinite ones.' But he is not impressed by this way of dealing with the problem: 'But surely the question that then needs to be asked is, How does one know that the Principle of Correspondence does not also hold for finite collections, but not for infinite ones? Here the mathematician can only say that it is simply defined as doing so.' [Morriston]
I mention the latter part of the quotation to give it context. But it's clear that revising EM is unnecessary for Craig's strategies. It seems to me that the real absurdities that result from HH-type scenarios involve the idea of contradictory results using prohibited inverse operations (we can get to Morriston's objections to this later). Addition or multiplication is counterintuitive on a metaphysical plane, but subtraction and division are what do the trick for me.
The other option (appeal to set-theoretical mathematics to motivate the mutual inclusivity of EM* and HP) also seems problematic. First, this seems to beg the question because it is assumed that one can successfully apply the Principle of Correspondence between the members of a mathematical set and concrete entities on a metaphysical plane. HH-type scenarios are designed to block the broad logical possibility of accomplishing such a correspondence. Second, as intimated above, EM and HP only seem to be applicable for finite collections, despite Cantor's animadversions of such a restriction. Hence, it's not that EM and HP conflict; they don't on a finite plane. It's that they don't apply on an infinite plane, whether mathematical or metaphysical.
Therefore, I will opt for rejecting HP and EM on an infinite plane and coaxing out HH-type absurdities, not on the basis of alleged absurdities arising out of the application of HP and EM on an infinite plane, nor on the basis of adopting pre-theoretical notions illicitly in the mathematical realm (motivating the mutual inclusivity of HP and EM), but on the basis of the contradictory results that find their origin in the inverse operations of subtraction/division of infinite multitudes. In this way, you can escape clashes between pre-theoretical and set-theoretical ideas and quarantine the explanation for the mutual exclusivity of HP and EM to the realm of inapplicability to the infinite plane, rather than to any absurdity arising out of an illicit conflation of pre-theoretical and set-theoretical ideas on the basis of an alleged mutual exclusivity between HP and EM.
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