Re: Request for Review of ZF Inconsistency Proof

(This post is a part of the campaign to save logic on Usenet. You can
show your support by posting sense).

MoeBlee <jazzmobe@xxxxxxxxxxx> writes:

If I am recalling correctly about this, this is what I LIKE about
his book: he gives theorems about levels (from some axioms that are
not the usual but turn out to provide an equivalent axiomatization
as the usual?) (except as you mention, allowing urelements?) to the
usual treatments.

Potter does base his development on levels, yes, and obtains Zermelo
set theory with ur-elements. The justification for the various axioms
of set theory on basis of the idea of sets being "built" in stages is
carefully considered, though in a style that, for some reason or
other, I found not quite to my liking.

One of my favourite observations that people usually find either
glaringly obvious or fail to appreciate entirely can also be found in
Potter's book. There is an important difference between the
transfinite recursion scheme

If G is a definable class function, there is a unique definable class
function F such that, provably in ZFC, F(alpha) = G(F restricted to
alpha) for all ordinals alpha

and the reflection scheme

For every P in the language of set theory, there is, provably in ZFC
an alpha, s.t P <--> (V_alpha |= P)

In the first case, the argument we give does not in any way depend on
the definability of G and F -- the only reason we mention definability
is that we're working in the language of set theory and have no class
quantification at our disposal; but with the reflection scheme, we
have to carry out an induction on the complexity of P, that is, it
does matter that P is a statement about sets expressible in the
language of set theory.

In _Reflection Principles, Large Cardinals, Elementary Embeddings_[1]
Reinhardt writes, talking about obtaining (small) large cardinals by
reflection:

Utilizing informal criteria to judge when the passage from "for all
definable F" to "for all F" is acceptable, one can in this way obtain
all the Mahlo cardinals. (The informal step is in each case analogous
to the passage from "V satisfies replacement for definable functions"
to "V is inaccessible.")

This passage is somewhat puzzling, since no "informal criteria"
suggest themselves that would allow one to ever pass from "V satisfies
replacement for definable functions" to "V is inaccessible". Rather,
the only reason anyone would think "V satisfies replacement for
definable functions" is that they were already convinced replacement
holds for whatever functions there happen to be, and, accepting the
legitimacy of functions definable in the language of set theory,
concluding on that basis that, in particular, replacement holds for
definable functions. (Of course, it is possible, perhaps even likely,
this is just what Reinhardt has in mind with "informal criteria").

The distinctions at issue can be captured formally, for example by
introducing class constants that effectively act as variables: no
properties of these class constants are stipulated, and thus the only
properties that of them we can use in proofs are that they are
classes. In addition, we add a rule of inference allowing a class
constant to be replaced with an open formula. The transfinite
recursion theorem can then be formulated as: G is a function -->
F(alpha,y) <--> <alpha,y> in G({beta,x} | F(beta,x) & beta < alpha}),
where F is a formula containing G. We get the instances of the usual
schematic form by replacing G with a formula in the language of set
theory. The reflection schema, and other schematic results that really
do only hold for definable properties, functions, statements
expressible in the language of set theory, have no such
reformulations; these results have no uniform proof, that is, the
structure of a proof of an instance of the scheme depends essentially
on the formula.


Footnotes:
[1] It is in this article the now famous "ultimate large cardinal
axiom", sometimes known as Reinhardt's axiom, is introduced, that there
is a non-trivial elementary embedding of the universe in itself. Kunen
showed that this axiom is, in presence of choice, inconsistent -- a
result occasionally referred to as the "Kunen inconsistency".

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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