Re: Principles of Induction in non well-founded set theories?

On Oct 4, 3:06 pm, Adam Burley <ajbur...@xxxxxxxxxxxxxx> wrote:
On Oct 4, 3:05 am, Adam Burley
<ajbur...@xxxxxxxxxxxxxx> wrote:

The paper is "The consistency of the axiom of
choice and of the generalized continuum hypothesis
with the axioms of set theory", 1940, p5, Axiom C1.
Reprinted in "Kurt Godel: Collected Works, Volume II
(Publications 1938-1974)", 1990, p38, Axiom C1.

Thanks. I think that might be the same (or virtually
the same?) as the
one in book form, of which I have only the first two
chapters (I'll
try to get to the library for more if needed though).

Yes, as far as I am aware, it is the same as the one in book form, but I think that this book form is very rare, I tried to get a hold of one once, but it was quite hard.

[...]
That said, I'm having difficulty proving that Godel's
axiom of
infinity entails the existence of an infinite set
(i.e., a set that is
not 1-1 with a natural number), not to mention the
existence of the
set of natural numbers, even though it is forcefully
intuitive to me
that his axiom does entail the existence of an
infinite set. (I guess
he does that in later chapters?)

I'm sorry, I don't quite understand you. Godel's axiom guarantees the existence of an infinite set. This is where infinite is defined as "equinumerous with one of its proper subsets".

That's Dedekind infinite, which implies plain infinite (where 'plain
infinite' is defined as 'not 1-1 with any natural number').

If you want to define infinite in terms of natural numbers, you will probably need induction to show it is equivalent to this definition.

The equivalence is shown by using the axiom of choice to show plain
infnite entails Dedekind infinite.

But what I'm not seeing offhand is how Godel's axiom of infinity
entails that there exists a plain infinite set (which is even weaker
than entailing there exists a Dedekind infinite set) let alone how his
axiom of infinity entails the existence of the set of natural numbers.
(But I haven't tried using the axiom of choice yet; but I don't think
choice should be needed anyway).

Ironically, Godel _does_ define infinite in terms of natural numbers, but he doesn't prove that w is infinite in this sense, to my knowledge.

It would follow from Dedekind infiniteness of w, given that he proves
that.

I'll try to make a point of looking it up at the library this weekend
(and hope it's not checked out this weekend!).

MoeBlee


.

Relevant Pages

  • Re: A puzzle for Cantorists
    ... Any universe of sets used as a model of ZF ... For example, I "believe" in the Axiom of Choice, ... comprehensive line of argument about why and how infinite sets are ... the hierarchy" is not mathematics. ...
    (sci.math)
  • Re: Inconsistency of the usual axioms of set theory
    ... exists a Dedekind-infinite set" implies the existence of N? ... one can prove that "there exists an infinite ... particular axiom of infinity ...
    (sci.math)
  • Re: Cantor Confusion
    ... existence of a set having a certain property just by referring to "the ... natural numbers and other infinite sets are present by naming N etc. ... model without an axiom of infinity securing its existence? ... theory is a set of sentences closed under entailment (cf., e.g., ...
    (sci.math)
  • Re: "Choosing" the choice relation
    ... deterministic infinite choice is not a problem. ... If determinism is what you crave, ... since you deny the existence of elements you can't specify, ... consider Godel's axiom of infinity (this is different from the ZFC axiom ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... There exists a set x such that 0 is a member of x and for any y, ... > That follows from your definition in the axiom of infinity. ... So omega is the least infinite ordinal. ... symbols back to primitives. ...
    (sci.math)