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ƧƿѦςɛ♏ѦਹѤʞProvably unprovable? <a href="https://mastodon.social/tags/WTF" class="mention hashtag" rel="tag">#WTF</a> :-) <a href="https://www.youtube.com/watch?v=0Le7NgS-wO0" target="_blank" rel="nofollow noopener" translate="no">https://www.youtube.com/watch?v=0Le7NgS-wO0</a> <a href="https://mastodon.social/tags/maths" class="mention hashtag" rel="tag">#maths</a> <a href="https://mastodon.social/tags/mathematics" class="mention hashtag" rel="tag">#mathematics</a> <a href="https://mastodon.social/tags/numbers" class="mention hashtag" rel="tag">#numbers</a> <a href="https://mastodon.social/tags/Goodstein" class="mention hashtag" rel="tag">#Goodstein</a> <a href="https://mastodon.social/tags/Peano" class="mention hashtag" rel="tag">#Peano</a> <a href="https://mastodon.social/tags/G%C3%B6del" class="mention hashtag" rel="tag">#Gödel</a> <a href="https://mastodon.social/tags/sequence" class="mention hashtag" rel="tag">#sequence</a>

Apuntes de cienciaMi hijo (13) me ha preguntado "Si 1=2, ¿cuánto vale 1+1+1+1?" y yo he aprovechado para hablarle de los axiomas de <a href="https://astrodon.social/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> 😋 ➡️<a href="https://es.wikipedia.org/wiki/Axiomas_de_Peano" rel="nofollow noopener" target="_blank">https://es.wikipedia.org/wiki/Axiomas_de_Peano</a>

Ian Douglas Scottfn main() { println!("{}", <S<S<S<S<Z>>>> as Nat>::Fact::INT); }<a href="https://fosstodon.org/tags/peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#peano</a>

SDF.ORGgosper's graph of a space filling function painting successive range points with gradient color <a href="https://mastodon.sdf.org/tags/hack" class="mention hashtag" rel="nofollow noopener" target="_blank">#hack</a> <a href="https://mastodon.sdf.org/tags/math" class="mention hashtag" rel="nofollow noopener" target="_blank">#math</a> <a href="https://mastodon.sdf.org/tags/peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#peano</a> <a href="https://mastodon.sdf.org/tags/lisp" class="mention hashtag" rel="nofollow noopener" target="_blank">#lisp</a> <a href="https://mastodon.sdf.org/tags/mandlebrot" class="mention hashtag" rel="nofollow noopener" target="_blank">#mandlebrot</a>

xameerSo which is more natural? Shapes which we see in nature or the natural numbers, which we need to define them in full? How could <a href="https://ioc.exchange/tags/peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#peano</a> just assume that 0 is a natural number ? What is a shape with zero sides? Is infinity a natural number too? What tells N from R?

xameer- In <a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> arithmetic, second-order arithmetic and related systems, and indeed in most (not necessarily formal) mathematical treatments of the well-ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic

xameer- strengthened finite <a href="https://ioc.exchange/tags/Ramsey" class="mention hashtag" rel="nofollow noopener" target="_blank">#Ramsey</a> theorem, is true, but not provable in Peano arithmetic. This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in <a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> arithmetic <a href="https://ioc.exchange/tags/godel" class="mention hashtag" rel="nofollow noopener" target="_blank">#godel</a>

xameer- Relation algebra algebraizes part of fol consisting of formulas having no atomic formula lying in scope of >3 quantifiers. Enough , for <a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> arithmetic , axiomatic set theory <a href="https://ioc.exchange/tags/ZFC" class="mention hashtag" rel="nofollow noopener" target="_blank">#ZFC</a>; hence relation algebra, unlike PFL, is incompletable

xameer<a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> and Mario Pieri used the expression motion for the congruence of point pairs

xameerHydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time. Just like for <a href="https://ioc.exchange/tags/Goodstein" class="mention hashtag" rel="nofollow noopener" target="_blank">#Goodstein</a> sequences, Kirby and Paris showed that it cannot be proven in <a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> arithmetic alone

xameerSince <a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> arithmetic cannot prove its own consistency by <a href="https://ioc.exchange/tags/G%C3%B6del" class="mention hashtag" rel="nofollow noopener" target="_blank">#Gödel</a>'s second incompleteness theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem.

xameerthere is an integer n such that if there is a sequence of rooted trees T1, T2, ..., Tn st Tk has at most k+10 vertices, then some tree can be <a href="https://ioc.exchange/tags/homeomorphically" class="mention hashtag" rel="nofollow noopener" target="_blank">#homeomorphically</a> embedded in a later one" is provable in <a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> arithmetic, but shortest proof > A(1000), where A(0)=1 and A(n+1)=2A(n)

xameerGödel: any <a href="https://ioc.exchange/tags/recursive" class="mention hashtag" rel="nofollow noopener" target="_blank">#recursive</a> system that is sufficiently powerful, such as <a href="https://ioc.exchange/tags/Peano" class="mention hashtag" rel="nofollow noopener" target="_blank">#Peano</a> arithmetic, cannot be both consistent and syntactically complete.

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