🚀 go-pugleaf

In article <TORKEL.92Nov19201017@lludd.sics.se>, torkel@sics.se (Torkel Franzen) writes:
> In article <363@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:
> 
>    >I think we can learn a lot more
>    >about the notion of a real number from such research than we can
>    >from proving theorems related to the continuum hypothesis.
> 
>   I have no objection to your inventing whatever mathematics you are
> capable of, on the basis of any ideas whatever. My negative comments
> are directed only at the notion that results in logic "show" that second
> order non-finitary concepts are not well-defined, or that we can do
> without them.

I have not claimed that these concepts are either not well defined or that
results in logic show we can do without them. My objection is a philosophical
one and centers on the assumption that mathematics requires quantification
over the reals be based on non-finitary concepts. You cannot define
the ordinal of the recursive ordinals without quantifying over the reals.
This is an important and useful concept. Does this mean that we must accept
that the set of all reals as a completed infinite totality? I do not think so.
I think the notion of an arbitrary path in a recursively enumerable tree is
meaningful even though I think the set of all reals is not.

Paul Budnik