WAtch out, Liz! you are getting close to ask about PRIME NUMBERS, what may mean a totally different trap! John M
On Mon, Mar 16, 2015 at 6:33 PM, LizR <[email protected]> wrote: > My apologies obviously you did mean finite. > > This is very interesting although probably too much for my brain at the > moment. > > What is all the stuff about S(S(0)) and {}, {{}}, etc? Doesn't that define > finite numbers? > > > On 17 March 2015 at 05:39, Bruno Marchal <[email protected]> wrote: > >> >> On 15 Mar 2015, at 21:29, meekerdb wrote: >> >> On 3/15/2015 10:03 AM, Bruno Marchal wrote: >> >> We cannot define the notion of finite number >> >> >> This will make it very difficult to interpret the output of your computer. >> >> >> I guess you are joking. >> >> In case you are serious, you really should study a good book on logic. >> >> Machines can handle many things that they cannot define. >> >> To make my statement more precise, it means that we cannot build a theory >> having all natural numbers and only the natural numbers as model, by using >> first order logic. In fact no theory of any finite things can be formalized >> in first order logic. There is no first order axiomatization of finite >> group theory, of finite field, etc. There are good theories, even first >> order theories, but they have infinite models. >> >> We can formalized finiteness in ... second order logic. But this is a >> treachery because this use the notion of finiteness (in explicit or >> implicit way). >> >> That is the root of the failure of logicism. Not only we have to assume >> the natural numbers and they additive and multiplicative structure, (if we >> want use them), but we can't interpret them categorically or univocally. It >> is a strange world where it can be consistent for a machine to be >> inconsistent. >> >> What I really meant was: we cannot define the notion of number without >> using the notion of finite number. >> You might try, as a game to define natural number without using the >> notion, like if explaining them to someone who does not grasp them at all >> (if you can imagine that). >> >> You might say I is a number, and: if x is a number, then Ix is a number. >> The difficulty is in avoiding the person believe that IIIII... become a >> number, with a variety of meaning for IIII... >> >> Bruno >> >> >> >> >> >> Brent >> >> -- >> 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 [email protected]. >> To post to this group, send email to [email protected]. >> Visit this group at http://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> >> >> http://iridia.ulb.ac.be/~marchal/ >> >> >> >> -- >> 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 [email protected]. >> To post to this group, send email to [email protected]. >> Visit this group at http://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> > > -- > 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 [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > -- 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
