All Questions
15 questions
43
votes
1
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2k
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Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?
By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
We are considering a ...
- 32.5k
22
votes
5
answers
1k
views
What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?
I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic.
Definitions. ...
- 236k
18
votes
1
answer
3k
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Existence of a model of ZFC in which the natural numbers are really the natural numbers
I know that, from compactness theorem, one can prove that there are models of first order arithmetic in which there is some "number" which is not a successor of zero, in the sense that it is strictly ...
- 1,514
16
votes
2
answers
1k
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How special is first-order $\mathsf{PA}$?
This is a modified version of a question which was asked and bountied at MSE without success.
Below, "$\mathsf{PA}$" refers to first-order Peano arithmetic.
There are various "...
- 21.2k
12
votes
2
answers
1k
views
Trouble with models of PA and ZFC
I have a big trouble in my mind, here is my false reasoning:
The Goodstein's theorem is undecidable in (first order) Peano Arithmetic.
There exist a non standard model N of PA where the Goodstein's ...
- 129
11
votes
2
answers
379
views
Can singular long models require less than PA?
Say that a long model is an $\mathfrak{A}\models\mathsf{I\Sigma_1}$ such that $\mathfrak{A}$ has strictly greater cardinality than each of its proper initial segments (in the case $\vert\mathfrak{A}\...
- 21.2k
9
votes
4
answers
3k
views
Incompleteness and nonstandard models of arithmetic
The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome.
Reading Peter Smith's "Gödel Without (Too Many) Tears",...
- 1,465
6
votes
1
answer
382
views
Formal systems needed to formalize relative independence results
We know that Con(ZF) implies Con(ZFC+GCH), Con(ZF+neg(AC)) and Con(ZFC+neg(CH)). But what are some weak theories in which these relative independence results are provable? In particular, are they ...
- 4,985
5
votes
3
answers
1k
views
Are there first-order statements that second order PA proves that first order PA does not?
Are there first-order statements that second order PA proves that first order PA does not? Is this known one way or the other? Could you share an example? (edit: to clarify, by 'second order PA' I don'...
- 117
5
votes
0
answers
317
views
$\Sigma_n$-complete sets in the Levy hierarchy
Recall that a set $A \subseteq \mathbb N$ is (many-one, Turing) $\Sigma_n$-complete if it's $\Sigma_n$ and any other $\Sigma_n$ set (many-one, Turing) reduces to it. This definition actually makes ...
- 1,882
4
votes
1
answer
515
views
Truth Values of Statements in non-standard models
Excuse me, if the question sounds too naive.
Non-standard models of PA will have statements of non-standard lengths, basically infinite. And it is also true that every statement of a theory will have ...
- 1
3
votes
0
answers
301
views
What does second order set theory give us that is new?
There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here.
Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, ...
- 18.7k
2
votes
3
answers
552
views
Generalizations of PA and its standard and non-standard models
Consider Peano's axioms — in its first-order version and without addition and multiplication — with its single injective function $S$:
$(\forall x) \neg Sx = 0$
$\Big(\phi(0)\ \ \&\ \...
- 9,720
2
votes
2
answers
1k
views
Are there non-commutative models of arithmetic which have a prime number structure?
Peano Arithmetic (PA) models have a prime number structure and commutativity of addition and multiplication. Presburger arithmetic (PrA) models of arithmetic have addition without multiplication and ...
user10290
1
vote
0
answers
89
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About synonymy relationships around these two theories?
The following question is about patterns of synonymy relationships around two theories, $T^+$ and $\sf PA$.
For purposes of self inclusiveness I'll re-iterate $T$ and its extensions.
$\textbf{Logic:}$ ...
- 11.3k