All Questions
16 questions
12
votes
4
answers
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Is this theory synonymous with PA?
Language: Mono-sorted first order logic with equality.
Extralogical Primitives: $<, \in$
Define: $x \leq y \iff x < y \lor x=y$
$\textbf{Well ordering: }\\\textit{Transitive:} \ x < y \land ...
- 11.3k
42
votes
7
answers
3k
views
How would one even begin to try to prove that a simple number-theoretic statement is undecidable?
This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...
- 29k
22
votes
5
answers
1k
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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
15
votes
5
answers
2k
views
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(...
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43
votes
1
answer
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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
32
votes
2
answers
3k
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Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?
There are many interpretations of arithmetic in set theory. The
Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor:
$$0=\{\ \}$$
$$1=\{0\}$$
...
- 236k
29
votes
10
answers
4k
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Defining the standard model of PA so that a space alien could understand
First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on ...
- 18.7k
1
vote
1
answer
313
views
What is the set theory synonymous with this order-set theory?
Let $ T$ be a theory written in Mono-sorted first order logic with equality, with extralogical primitives: $<, \in$.
Define: $x \leq y \iff x < y \lor x=y$
Axioms:
$\textbf{Well ordering: }\\\...
- 11.3k
27
votes
5
answers
4k
views
What is induction up to $\varepsilon_0$?
This is a question asked out of curiosity, and because I can't understand the Wikipedia page.
I have often been told that PA cannot prove the validity of induction up to $\varepsilon_0$, which has ...
- 156k
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 "...
- 20.5k
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
8
votes
0
answers
1k
views
What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
- 16.6k
6
votes
1
answer
727
views
What is the consistency strength of this theory?
Language: first-order logic
Primitives: $=, S, \in $ (the first denotes identity, the second denotes “is a successor of”, and the third denotes membership relation).
Axioms: those of identity ...
- 11.3k
6
votes
3
answers
1k
views
Set theory inside arithmetics via the Ackermann yoga
Among the basic results of logic which, simple as they are, never fail to intrigue me, is Ackermann's interpretation of ZF-Infinity in PA (see for refs this MO question and here for an excellent ...
- 7,853
2
votes
1
answer
142
views
Does $WKL_0$ plus CON(PA+X) give a binary tree model of PA+X?
In the context of reverse mathematics $WKL_0$ is considered equivalent to Gödel's completeness theorem over $RCA_0$. Does this mean that e.g. $WKL_0$ plus the consistency statement CON(PA+X) gives a ...
- 3,574
1
vote
1
answer
213
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Is there an effective way to generalize this approach of affinely extending the number line?
The general approach here is a follow up of the approach outlined in a prior posting on extending the projectively extended real line. In particular arithmetic operators break down to ternery ...
- 11.3k