Correct me if I'm wrong but my understanding is that sets and membership cannot be defined in terms of a more primary mathematical concept. Functions can be defined in terms of this primitive called sets. Numbers are sets; natural numbers are defined directly in terms of sets (via the Von Neumann approach) and every more complicated number set can be defined in terms of the previous type of number set all the way up to real numbers, complex numbers, and nonstandard number sets. The only type of number I am not sure how they can be seen as sets is that of surreal numbers described by Conway I believe. I don't know much about surreal numbers.
Yes, this approach necessitates the existence of sets and membership. There probably are other ways to define numbers but all properties that we want numbers to have can come from how they are defined in terms of sets. In other words, the set theoretical description of various number sets is sufficient. Kronecker said "God made the integers; all else is the work of man." I would amend that to say God made sets (and membership); all else is the work of man. On Thu, Sep 6, 2012 at 7:45 AM, Stephen P. King <[email protected]>wrote: > Dear Brian, > > "can be defined ..." implies the necessary existence of something or > process or whatever that does the act of defining the set. Truth values do > not do this, btw. Sets are collections defined in terms of functions, but > numbers in-themselves are not those functions.. Unless you are considering > some other ideas of what sets are... If we are going to think of set as > having ontological primacy we have to have a notion of a set that does not > need a membership function. > > > > On 9/6/2012 10:28 AM, Brian Tenneson wrote: > > All numbers can be defined in terms of sets. The question becomes this: > do sets have ontological primacy relative to mankind or are sets invented > or created by mankind? > > On Thu, Sep 6, 2012 at 5:11 AM, Roger Clough <[email protected]> wrote: > >> Hi Stephen P. King >> >> >> Yes, of course, but I wanted a more obvious, dramatic example. >> The philosophy of mathematics says something like the numbers >> belong to a static or eternal world, change itself is a property of >> geometry. >> Numbers and geometry thus belong to the platonic world, >> which is forbidden or at least not consistent with the philosophy >> of materialism, IMHO. >> >> If numbers are platonic, I wonder what the presumably materialist >> Steven Hawkings has to say about their origin in his recent >> book on numbers. >> >> >> >> Roger Clough, [email protected] >> 9/6/2012 >> Leibniz would say, "If there's no God, we'd have to invent him >> so that everything could function." >> >> ----- Receiving the following content ----- >> *From:* Stephen P. King <[email protected]> >> *Receiver:* everything-list <[email protected]> >> *Time:* 2012-09-06, 07:53:18 >> *Subject:* Re: Could we have invented the prime numbers ? >> >> Dear Roger, >> >> Could the mere possibility of being a number (without the specificity >> of which one) be considered to be "there from the beginning"? >> >> > > > -- > Onward! > > Stephen > http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
