Torgny Tholerus wrote: > Brian Tenneson skrev: > >> This is a denial of the axiom of infinity. I think a foundational set >> theorist might agree that it is impossible to -construct- an infinite >> set from scratch which is why they use the axiom of infinity. >> People are free to deny axioms, of course, though the result will not >> be like ZFC set theory. The denial of axiom of foundation is one I've >> come across; I've never met anyone who denies the axiom of infinity. >> >> For me it is strange that the following statement is false: every >> natural number has a natural number successor. To me it seems quite >> arbitrary for the ultrafinitist's statement: every natural number has >> a natural number successor UNTIL we reach some natural number which >> does not have a natural number successor. I'm left wondering what the >> largest ultrafinist's number is. >> > > It is impossible to lock a box, and quickly throw the key inside the box > before you lock it. > It is impossible to create a set and put the set itself inside the set, > i.e. no set can contain itself. > It is impossible to create a set where the successor of every element is > inside the set, there must always be an element where the successor of > that element is outside the set. >
Depends on how you define "successor". Brent > What the largest number is depends on how you define "natural number". > One possible definition is that N contains all explicit numbers > expressed by a human being, or will be expressed by a human being in the > future. Amongst all those explicit numbers there will be one that is > the largest. But this "largest number" is not an explicit number. > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
