All Questions
Tagged with peano-axioms set-theory
80
questions
1
vote
0
answers
82
views
Godel's incompleteness theorem: Question about effective axiomatization
I am studying Godel incompleteness theorems and I am struggling with the definition of effective axiomatization.
From Wikipedia:
A formal system is said to be effectively axiomatized (also called
...
- 2,452
0
votes
3
answers
241
views
Allegedly: the existence of a natural number and successors does not imply, without the Axiom of Infinity, the existence of an infinite set.
The Claim:
From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:
[T]he existence of natural numbers and the fact that given a natural ...
- 45.7k
0
votes
1
answer
117
views
Proving the Weak Goodstein Theorem within $\mathsf{PA}$
In
Cichon, E. A., A short proof of two recently discovered independence results using recursion theoretic methods, Proc. Am. Math. Soc. 87, 704-706 (1983). ZBL0512.03028.
the following process is ...
- 4,432
1
vote
2
answers
104
views
Model Theory in the Language of Peano Arithmetic
Most introductory textbooks on model theory establish the theory based on the ZF set theory (e.g. [1]). In particular, a structure is defined to be a 4-tuple of sets, and so on. In [2], I came to ...
- 1,822
6
votes
2
answers
887
views
Formally how do we view finite sets
This might be silly, but I have been thinking about how we would work with finite sets very formally.
So, $\{1,2,3,\cdots,n\} = \{k \in \mathbb{Z}^+ \mid k \leq n\}$ gives a representee for which any ...
4
votes
1
answer
308
views
Is "standard $\mathbb{N}$" in fact not "fully formalizable"?
Note: "Update" at the end of this question hopefully summarizes/clarifies the original language (original text left in place for context).
Philosophical Preface: For the purposes of this ...
- 1,024
1
vote
1
answer
83
views
Does the property $T\vdash Pvbl_T(\ulcorner \sigma \urcorner) \implies T\vdash \sigma$ apply to set theories?
I know from other posts that $PA\vdash Pvbl_{PA}(\ulcorner \sigma \urcorner ) \implies PA\vdash \sigma$ and this applies to other extensions/restrictions of PA as well. Does it also apply to set ...
- 855
0
votes
1
answer
163
views
Analysis I, can Tao's construction of the integers be further simplified?
In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers:
In the language of set theory, what we are doing here is starting with the ...
- 11
1
vote
1
answer
115
views
Are the natural numbers definable in ZFC-Inf
While reading up on the incompleteness theorems on this page, I came upon the statement that the incompletess theorems hold in ZFC-Inf. According to the page ZFC-Inf is 'proof-theoretically equivalent'...
- 149
0
votes
1
answer
544
views
Proving $\Sigma_1$ Completeness of Peano Arithmetic
Can we prove that PA is $\Sigma_1$-complete with PA, or do we need to use a stronger theory like ZFC?
In either case, what does the proof look like? What stops ZFC from being $\Sigma_1$-complete?
user1170736
1
vote
1
answer
183
views
What ZF can do and Peano's axiom cannot.
I am interested in how much math can be done from Peano's axioms and what can't.
What is there in the mathematics done with ZF that cannot be done with Peano's axioms?
- 390
4
votes
1
answer
348
views
What is the mathematical definition of "standard arithmetic/standard natural numbers"?
As a consequence of Godel's incompleteness theorem, no axiomatic system can be the definition of standard natural numbers because any axiomatic system of arithmetic will always be satisfied by non-...
- 1,437
1
vote
1
answer
288
views
Can Goodstein's theorem be false in some model of Peano arithmetic? How?
I read that Goodstein's theorem is not provable in Peano arithmetic, but is provable in ZFC.
Now, using Godel's completeness theorem, we can say that, since Goodstein's theorem is not provable in ...
- 1,437
0
votes
2
answers
134
views
Peano structure ordering and the recursion theorem circular definitions
Suppose $(P, 0, S)$ is a Peano structure. I am trying to prove the Recursion theorem* and I'm mixed up as to if the recursion theorem needs to be proven first or the order $<$ needs to be defined ...
- 1,461
0
votes
0
answers
41
views
Peano' s Systems
Lets consider $(A,g,a_{0})$ , where $A$ is a set, $g$ is a function and $a_{0}$ is the first element of $A$. If $(A,g,a_{0})$ follows all the Peano's axioms, than we can find an isomorphism f between $...
- 153