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Tagged with peano-axioms induction
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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 ...
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How to justify why succession and addition cannot be circularly defined like this?
I am reading Tao's Analysis I, in which he states:
One
may be tempted to [define the successor of $n$ as] $n + 1$. . . but this would introduce a circularity in our foundations, since the notion of ...
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Why is addition not completely defined here?
Say for the natural numbers, we define addition this way:
$0 + 0 = 0$, and if $n+m = x$, then $S(n) + m = n+S(m) = S(x) $
Say we have the regular Peano axioms, except we delete the axiom of ...
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Is it circular to include reachability from $0$ like this as a Peano axiom?
I am wondering whether it makes logical and semantic sense to include, as an axiom to define the natural numbers, that "every natural number is either $0$ or the result of potentially repeated ...
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Can we modify the Peano axioms like this? [closed]
I am wondering if the following modifications of the Peano axioms result in a set of axioms equivalent to the Peano axioms, in the sense that any set of numbers satisfies these modified axioms if and ...
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Proving that the set of non-negative half-integers satisfies Peano's axioms
I postulate that the following set $\{0,0.5,1,1.5,...\}$ represents the natural numbers. Of course, intuitively, this isn't true. But let me try to show this using Peano's axioms. I'll first define ...
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Help me check my proof of the cancellation law for natural numbers (without trichotomy)
can you guys help me check the fleshed out logic of 'my' proof of the cancellation law for the natural numbers? It's in Peano's system of the natural numbers with the recursive definitions of addition ...
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Do $P(0)$ and $P(n)\implies P(n+1)$ yield $P(5)$ without an axiom of induction?
As I understand it, Peano arithmetic needs the axiom of induction to prevent non-standard models of the natural numbers.
Given $P(0)$ and $P(n)\implies P(n+1) \forall n\in \mathbb{N}$ I can apply ...
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Irregular Induction Theorem for $\mathbb{N}\times\mathbb{N}$
I am trying to prove this irregular induction theorem that would help prove a recursion theorem I am working on.
Can you help? Here is the theorem:
$\forall X (\forall x \in \mathbb{N} (\langle x,0 \...
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How to prove natural number addition using induction? [duplicate]
I am a self learner so excuse me if I am asking a seemingly easy question , But I ve been stuck at this point for couple of days , I think I understand mathematical induction and what the author is ...
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Proof of Recursive definition, Analysis 1 by Terence Tao.
I got Proof of a proposition regarding recursive definitions (from Terence Tao's Analysis I)
Here i understood that what tao done in the proof. But still i have some confusion.
Question: Why ...
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What does it mean that we need $𝜖_0$ induction to prove PA consistency?
I have started to learn about Peano Arithmetic, and also about ordinals.
In particular, I have seen that the Goodstein theorem is an example of a statement that can be expressed in PA but that ...
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Peano axiom of induction with "no junk"
In this Wikipedia treatment of Peano Axioms, if you go down to the first picture you'll see a circle of dominoes and a straight line of dominoes:
The caption says the straight line of dominoes
The ...
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Proof of the Principle of Backwards Induction
I'm trying to prove the following proposition, where "++" denotes the successor function (i.e., 2++ = S(2) = 3).
Let $n$ be a natural number, and let $P(m)$ be a property pertaining
to the ...
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Successor axiom in Robinson arithmetic
The successor axioms of Robinson Arithmetic (Q) are:
$\forall x\,(Sx\neq0)$
$\forall x\forall y\,[(Sx=Sy)\rightarrow x=y]$
$\forall y\,[y=0\;\lor\;\exists x\,(Sx=y)]$
Note that 3. differs from the ...
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