All Questions
Tagged with peano-axioms nonstandard-models
46
questions
1
vote
0
answers
95
views
Gödels incompleteness theorem false for natural numbers
Do I understand it correctly that in an axiomatic system that includes the natual numbers defined by using Peano's axiom, where the 9th axiom (induction) is formulated using 2nd-order logic, then
...
- 119
3
votes
3
answers
195
views
If Goldbach's conjecture G is undecidable in PA, then can we prove $\mathbb N\models G$?
Suppose Goldbach's conjecture $G$ is undecidable in first-order Peano arithmetic, $\sf{PA}$. That would mean there are models in which $G$ is true and other in which it is false. But intuitively, this ...
- 6,672
7
votes
1
answer
151
views
Extending a Model of $ T + \operatorname {Con} ( T ) $ to a model of $ T + \neg \operatorname {Con} ( T ) $
Let $ T $ be a recursively axiomatizable extension of $ \mathsf {PA} $ and $ \mathfrak M $ be a model of $ T + \operatorname {Con} ( T ) $. Is it true that there must exist a model $ \mathfrak N $ ...
- 6,766
0
votes
1
answer
43
views
standard model of $\mathbb{N}$ and true $\Pi^0_1$ sentences
Does the fact that provability of some true $\Pi^0_1$ sentences is equivalent to the existence of particular (I do not know from the top of my head to which ones, does someone know how they are called)...
- 3,978
4
votes
1
answer
184
views
Why does this nonstandard model of Robinson Arithmetic fail to be a model of PA?
As talked about in this question, there is a nonstandard model of Robinson arithmetic of the form:
$z_0 + z_1\omega + z_2\omega^2 + z_3\omega^3 + ... + z_n\omega^n$
in which $\omega$ is a formal ...
- 8,000
3
votes
2
answers
243
views
(Request for) simple constructive proof of existence of nonstandard model of PA
I know of two straightforward nonconstructive proofs of the existence of nonstandard models of arithmetic.
By the existence of the standard model of PA, PA is satisfiable and has an infinite model. ...
- 11.9k
0
votes
0
answers
139
views
How does second-order arithmetic rule out non-standard numbers?
According to 1 there are axioms of second-order arithmetic which categorically characterize the natural numbers up to isomorphism. The axioms are:
$$\forall x\,\neg(x+1=0)$$
$$\forall x\,\forall y(x+1=...
- 402
0
votes
1
answer
128
views
Is this an example of the incompleteness of first-order PA?
Although the usual natural numbers satisfy the axioms of PA, there are
other models as well (called "non-standard models"); the compactness
theorem implies that the existence of nonstandard ...
- 14.8k
1
vote
1
answer
44
views
A non-standard model of PA with total antisymmetric order without induction
I'm looking for a model of PA without induction whose order relation total and Anti-symmetric.
to be specific, satisfiying:
$\forall x \ (0 \neq S ( x ))$
$\forall x, y \ (S( x ) = S( y ) \...
- 2,225
1
vote
2
answers
144
views
Is there a first order formula that is satisfied by $\mathbb{N}$, but not by any other models of Peano axioms?
Is there a first order formula that is satisfied by $\mathbb{N}$, but not by any other models of Peano axioms? For example, is there a formula that expresses "there is an element that is greater ...
- 1,595
3
votes
1
answer
111
views
Extension of the Paris-Harrington principle
Let $[m,n]$ denote the set $\{m,m+1, ... ,n-1,n\}$. $X \to (k)^n_c$ means that whenever $f: [X]^n \to c$ there is a subset $H \subset X$ with cardinality $k$ such that $f$ is constant on $[H]^n$ (The ...
- 1,718
0
votes
1
answer
153
views
Set-up for the Paris-Harrington Theorem
In his book "Models of Peano Arithmetic" Kaye proves the Paris-Harrington Theorem. He starts off by introducing a "simplification" then proves the short Lemma 14.11 about it (see ...
- 1,718
1
vote
0
answers
63
views
Motivation of indicator construction in Kaye
Kaye says the following in his book about models of $\textbf{PA}$ on p. 198:
I have no clue what motivates the definition of $f_n(x, y)$. The reference made to Propositions 14.1, 14.2 don't help me ...
- 457
0
votes
1
answer
74
views
$\text{Sat}_{\Delta_0}(x, y)$ is $\Delta_1(\textbf{PA})$ (Kaye's book)
In Kaye's "Models of Peano Arithmetic" in chapter 9 on satisfaction a lot of effort is expanded to prove that $\text{Sat}_{\Delta_0}(x, y)$, representing truth of $\Delta_0$-sentences, is a $...
- 1,718
10
votes
3
answers
784
views
How can induction work on non-standard natural numbers?
When we consider the Peano axioms minus the induction scheme, we can have strange, but still quite understandable models in which there are "parallel strands" of numbers, as I imagine in the ...
- 30.1k