All Questions
Tagged with peano-axioms second-order-logic
20
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Why is the material conditional treated like logical entailment in second order quantification? [closed]
According to this Wikipedia article the second order axiom of induction is: $$\forall P(P(0)\land \forall k(P(k) \to P(k+1)) \to \forall x(N(x) \to P(x))$$
Where N(x) means x is a natural number. That ...
- 301
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Why is does this first-order set of axioms NOT genetically define the natural numbers?
It is a theorem of model theory that any recursively enumerable set of axioms $\Gamma$ for number theory permit non-standard models. That is, if there is one model for $\Gamma$, then there are two ...
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A Peano system with an infinite initial segment
Let $T$ be a binary tree with the lexicographic order. And $f:T→T$ be the successor operation.
We denote the empty sequence in $T$ by $i$.
Also Suppose that:
For all $X⊂T$ if these two conditions hold:...
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The set of all finite sequences in RCA0
In the book "subsystems of second order arithmetic", page 68, Simpson claimed that the set of all codes of finite sequences (denoted by "Seq") exists by sigma 0-0 comprehension. I ...
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In RCA0, Prove that for all n, fⁿ(i) exists
I wanted to prove in RCA0 that:
If f:A→A be a function and i∈A, then for all n, fⁿ(i) exists.
To reach that, I proved another theorem. I wrote my proof but I'm not sure it's rigorous. If you read it, ...
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1
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62
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Does Z₂ Prove the iteration theorem?
iteration theorem:
Let (ℕ,0,S) be the structure of natural numbers. And let W be an arbitrary set and c be an arbitrary member of W and g be a function from W into W. Then, there is a unique function ...
1
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What is the importance of Peano categoricity?
We know that Dedekind in 1888 proved that second order peano arithmetic(PA2) is categorical. My question is, why is it important? Does it have any mathematical, philosophical or foundamental ...
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Peano categoricity is equivalent to weak konig lemma
Peano categoricity (PC) says that:
Every model for second order peano system is isomorphic to standard model.
i.e PC says that every peano system such as
(A, f, i) is isomorphic to (N, S, 0).
Simpson ...
4
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2
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Incompleteness theorem: Peano arithmetic vs. standard model of arithmetic
Context (from
https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic#From_the_incompleteness_theorems):
The incompleteness theorems show that a particular sentence G, the
Gödel sentence of ...
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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
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Formal or informal definability of the standard model of natural numbers
I started a discussion in comments to this answer, but it grew beyond what fits comments, so I'm promoting it to this separate question. Here is a recap:
$\Large\color{green}{\unicode{0x2BA9}}\,$ The ...
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What subsystem of second-order arithmetic proves the weak Godel’s Theorem?
Godel’s Incompleteness Theorem states that any omega-consistent recursive theory in the language of first-order arithmetic is incomplete. But there is a weaker version of Godel’s Theorem that is ...
- 10.3k
4
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Is the axiom of induction required for proving the first Gödel's incompleteness theorem?
I'm reading a book about the mathematical logic. In the 6.3 chapter of that book, a theory $Q$ is introduced that contains precisely these axioms:
$Q1: \forall x. (S(x) \not= 0)$
$Q2: \forall x,y. (...
- 410
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When can a statement in Second Order Logic be converted into a statement in First Order Logic
I was reading on first and second order logic (The first has quantifiers over individuals of the domain and the second can have quantifiers over predicates as well - see this question: Differentiating ...
- 437
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What is the omega-completion of $ACA_0$
The omega rule is an infinitary rule of logic which says that from $\phi(0),\phi(1),\phi(2),...$ you can infer $\forall n\phi(n)$. My question is, what is the theory $T$ obtained by adding the omega ...
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