5
votes
2answers
60 views

How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
2
votes
2answers
109 views

How can we define infinitary proofs?

In the first order logic the usual notion of a formal proof for a sentence $\sigma$ from a theory $T$ is a "finite" sequence ($<\omega$ - sequeance) of sentences which each one of them is a valid ...
1
vote
1answer
128 views

Can the ongoing need for a meta language be stopped by a loop?

As an afterthought to this question on sets in set theory, and more specifically to the observation that a (first-order) logic requires a meta-language to explain itself (i.e. there is already an ...
5
votes
5answers
588 views

Induction versus Natural Numbers

$0$ is finite. If $n$ is finite, then $n+1$ is finite. Hence, by induction, all numbers are finite. What is the catch?