On Mon, Jun 8, 2015 Bruce Kellett <bhkell...@optusnet.com.au> wrote:
> What axioms led to arithmetic? The Peano axioms. They were chosen because they are very simple and self evident. You need to be very conservative when picking axioms, for example we could just add the Goldbach Conjecture as an axiom, but then if a computer found a even number that was NOT the sum of 2 primes it would render all mathematical work done after the addition of the Goldbach axiom gibberish. Or take Zermelo–Fraenkel set theory (ZFC) and the Continuum Hypothesis which says that there is no infinite number greater than the number of integers but less than the number of Real Numbers; in 1940 Godel showed that ZFC cannot prove the Continuum Hypothesis to be incorrect, and in 1963 Paul Cohen showed that ZFC cannot prove the Continuum Hypothesis to be correct either. So ZFC has nothing to say about the Continuum Hypothesis one way or the other. You could just add an axiom to ZFC saying "the Continuum Hypothesis is true" but you could just as easily add "the Continuum Hypothesis is NOT true", so which one do you add? The problem is that neither of these axioms are simple and neither are self evident. > > Could one have chosen different axioms? It's never a good idea to change axioms unless somebody finds a set of axioms that are even simpler and even more self evident. John K Clark -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
