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1d
comment How to prove that a set is infinite iff it is Dedekind infinite?
Also, there are infnite sets $X$ such that there is no bijection from $\mathbb{N}$ to $X$. These are called uncountable. I don't know how much you already know, perhaps you already knew that.
1d
comment How to prove that a set is infinite iff it is Dedekind infinite?
If $X$ is countably infinite then of course there is a bijection $b : \mathbb{N} \to X$ (by definition of "countably infinite"), but the proof of Proposition 1 need not produce such a bijection: it could happen that the $i$ we get from the proof is not surjective, even though some other map $b : \mathbb{N} \to X$ is surjective.
Aug
28
comment Models of set theory
I wrote a shorter answer here as well.
Aug
28
answered Models of set theory
Aug
28
comment Models of set theory
Perhaps the gist of my answer was the last sentence: if you ask "why are model theorists justified in using sets?" then I ask back "why are number theorists justified in using numbers?"
Aug
28
comment Models of set theory
A similar question was asked on MathOverflow, which I answered at length: mathoverflow.net/questions/23060/set-theory-and-model-theory/… I recommend the answer :-)
Aug
28
comment How to prove that a set is infinite iff it is Dedekind infinite?
I am not saying you swept it under the rug. I am just saying that doing so creates more trouble in the long run than it's worth.
Aug
27
comment How to prove that a set is infinite iff it is Dedekind infinite?
I wrote down the proof as I would (and will) teach it to my first years. We'll see how it goes. Also, I think we need to be honest with the students about choice. If we sweep it under the carpet, how will they ever learn to detect it?
Aug
27
answered How to prove that a set is infinite iff it is Dedekind infinite?
Aug
27
comment How to prove that a set is infinite iff it is Dedekind infinite?
The proof that an infinite set contains a denumerable set is hiding an application of choice under the carpet, and so it is confusing. Furthermore, the set $A$ in that part of the proof is not what you think the set $A$ is.
Aug
27
comment How to prove that a set is infinite iff it is Dedekind infinite?
$A \subset X$ does not imply $|A| < |X|$, consider for example $X = \mathbb{N}$ and $A = \{2 n \mid n \in \mathbb{N}\}$.
Aug
27
comment Why is $0/0$ not $\Bbb R$?
You could also accept the answer by clicking on the check next to it :-)
Aug
26
answered Why is $0/0$ not $\Bbb R$?
Aug
11
revised What is so special about $a*b^{ -1}$ equivalence?
deleted 40 characters in body
Aug
11
answered What is so special about $a*b^{ -1}$ equivalence?
Aug
9
comment How is the Gödel's Completeness Theorem not a tautology?
Tarski's definition of a sentence being true is what you think of as a sentence being true. Just skip the word "Tarski".
Aug
7
comment D={ $ deg_T (A) | A \subseteq N$} Problem
No. Halting problem is the maximum degree of computably enumerable degrees, but you defined $D$ to be all degrees. You are confused about basic definitions.
Aug
6
comment D={ $ deg_T (A) | A \subseteq N$} Problem
I means 1 and 2 false.
Aug
6
answered D={ $ deg_T (A) | A \subseteq N$} Problem
Aug
3
answered Is it possible to prove that the encoding of existentials in System F is valid?