Re: Logarithm of transfinite numbers

Hi, Tony. Just wanted to jump in here. :-)

The point is that there is no finite n in N for which the set
achieves infinity.

If by "achieving infinity", you mean "is of infinite size", then you
are correct. Infinite sizes cannot be achieved by finite increments of
finite sizes. You must have either an infinite number of increments or
an increment of infinite size.

This is true for every finite natural, that it has this relation
to the set size.

You are correct, this is true for every finite natural.

So, this inductively proven property of the element in
relation to the set size has implications for the set size
as a whole.

No, it inductively proves it for all set sizes which are finite natural
numbers. It is not proven (nor is it even true) for sets of infinite
size.

As long as the elements are all finite, so is the set
size.

Untrue, as explained above. Although this proposition is does hold for
all sets of finite naturals which has a greatest element. This is the
operative difference between sets like {1,2,3,...,n} and sets like
{1,2,3,...}.

If at every point in the set the value of the element at that point is
equal to the count of elements up to that point, and if at every
point in the set the value of the element at that point is finite, then
at every point in the set the number of elements up to that point is
finite, so there is NO point in the set where the number of
elements up to that point is infinite.

You need to define what you mean by "point". If you mean stopping at
some finite natural n and looking at the subset {1,2,...,n}, then you
are correct, there is no "point" (subset up to some finite natural) at
which the set becomes infinite. Likewise, at no such "point" soes the
complement {n+1,n+2,...} ever becomes finite. To get to the infinite,
you must transcend finite "points".

Explain exactly where this argument fails.

Your argument fails when you define "point" as a finite delta from the
beginning, and then assuming that an infinite sized set must become
infinite at some such "point". Of course at no "point" does it happen
(as you have defined it), but of course it is still infinite ("points"
notwithstanding).

There isn't any largest finite, and so your set's not
well defined...
<snip>

Untrue. Set definition is not dependent upon a largest element or a
largest finite.

...but if you insist on talking about finiteness as part
of the definition of a set, then you're going to have
these problems.

Not at all. The Dedekind definition of finiteness simply says that any
set which can be placed into 1-1 correspondence with a proper subset of
itself is infinite; all other sets are finite. This definition
requires no understanding of natural numbers or greatest elements or
anything of the kind. With this definition, you can easily see that no
set of the form { 1,2,...,n } is ever infinite, whilst { 1,2,... }
clearly is.

Regards,

Jonathan Hoyle

.

Relevant Pages

  • Re: Logarithm of transfinite numbers
    ... or the value of the string I am discussing. ... There is no point at which any finite natural has an infinite ... They ARE the finite naturals. ... At which point has my abstraction differed from what you're ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Randy Poe said: ... Do any of your naturals achieve any infinite values? ... or the value of the string I am discussing. ... They ARE the finite naturals. ...
    (sci.math)
  • Re: infinity
    ... there is no exact size of the set. ... >>> not matter whether N is finite or infinite, ... >> resulting set suddenly only has half the number of members. ... Thus, for the set of finite naturals, which does have a smallest, to ...
    (sci.math)
  • Re: infinity
    ... >>means the set CANNOT be infinite. ... > "every finite natural has finitely many predecessors" ... > "The set of finite naturals is a finite set" ... Because if an element has an infinite number of successors, ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... then it cannot be an infinite set. ... That is easily shown as false as the set of finite naturals (which ... performing an infinite number of increments on a finite value?" ... correct though that the "number of unit intervals" is indeed Aleph_0, ...
    (sci.math)
Loading