Questions tagged [induction]
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8
questions
32
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Is PA consistent? do we know it?
1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs (...
7
votes
1
answer
484
views
Is $\in$-induction provable in first order Zermelo set theory?
Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which $\in$-induction fail?
I asked this ...
36
votes
3
answers
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the following inequality is true,but I can't prove it
The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...
23
votes
5
answers
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Is Cauchy induction used for proofs other than for AM–GM?
The proof by Cauchy induction of the arithmetic/geometric-mean inequality is well known. I am looking for a further theorem whose proof is much neater by this method than otherwise.
19
votes
8
answers
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Unconventional types of induction
Induction is one of the most common tools is mathematics, and everybody knows the ordinary induction and the strong induction. However, in some proofs induction is applied in an unexpected and elegant ...
12
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6
answers
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Induction vs. Strong Induction
Is there ever a practical difference between the notions induction and strong induction?
Edit: More to the point, does anything change if we take strong induction rather than induction in the Peano ...
9
votes
1
answer
932
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ERA, PRA, PA, transfinite induction and equivalences
I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist.
For instance I'm ...
1
vote
2
answers
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Does Regularity schema imply $\in$-induction when added to first order Zermelo set theory?
That $\in$-induction fails to be a theorem schema of first order Zermelo + Foundation (see here), then it appears that it is more eligible to replace axiom of Regularity (Foundation) by a Regularity ...