Re: Logarithm of transfinite numbers

In article <MPG.1e8b8971d2df98a698ab6e@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1e8b557d7e89a7b898ab62@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

If each "increment" has an infinite number of others following it, as is
the case, it does no matter how many or how few it has preceding it.

If no finite is infinitely greater in units than any other finite, then how
can
there be an infinite number of unit increments and successions after any of
them, when there is nothing infinitely after it in the set?

One first gets the set of naturals via the inductive axiom, one then
shows that the set produced by that axiom satisfies our mathematical
definition of an infinite set. That's how!


The set of naturals is characterized by the equality between element
count and value within the set.

The set of naturals is *characterized* by the Peano axioms, and in no
other way. Anything other than the properies stated in those axioms is
derived, not intrinsic.

If TO has something other than that in mind, it is NOT the set of
naturals.



Every set of naturals having a lagest element is Dedeking finite.
Every set of naturals, except the empty set, NOT having a largest
element is provably Dedekind infinite.

Even if it doesn't contain an infinite quantity of elements, yes.

So TO here claims infinite sets are not infinite? Not very bright, is he?


As Dedekind infiniteness for sets is the only mathematically valid
definition of infiniteness for sets, TO is full of crap.


Yes, I know, set theorists think they own the foundations of mathematics and
theirs is the only valid approach.

Their axioms and definitions rule. If TO wants to insist on a word, let
him use "uncountable" since it is already defined and conveys the
meaning he wants to convey.

For TO to insist on misusing a word already used for something else
instead of using a perfectly good word which conveys his precise meaning
correctly is the act of a troll. So TO is spelled TrOll.


That would be fine if their system worked

It works better that the system that TrOll has never been able to get
off the ground. There are logically impeccable proofs of the properties
which are claimed to follow from our system of axioms, whereas there are
no proofs and not even any axioms for TrOll's.


,
but it spews paradoxes and ridiculous conclusions as if that's its job. In
fact, I think people rather like it because it's so mysterious. Oooohh,
spooky!
Well, for my part, I expect real mathemtics to draw reasonable conclusions.



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".

But, if all of the elements in the set are finite, and in positions
within
the
set equal to their finite value, then there exists no element in the set
which
marks anything but a finite set

Does TO claim that a finite number must to mark the end of an
unboundedly increasing sequence? Stupidity squared!



There si no "end" to the unboundedly increasing sequence, but according to
you,
all values are finite, and as inductively proven, that means all element
indexes are finite as well, being equal to the value that they index.


You
do not have an infinite set of naturals in the quantitiative sense until
you
have infinite quantities in the set.

In that case, TO has no infinite sets at all.


The naturals may contains infinite values in my opinion, despite your
discomfort with the notion, and continuous dense sets are infinite within any
finite value range, so you're spewing garbage as usual.




Sure, with the Dedekind set-theoretic definition that's true, as I
said above, but


But nothing. Dedekind infiniteness is the only infiniteness we recognise
as having any set theoretic meaning. TO's attempts to confuse the issue
are mere trolldom.


Your refusal to recognize any other ideas is evidence of your own fascism.
Relax.
.

Relevant Pages

  • Re: infinity
    ... >> Not while discussing the consequences of the axioms as they ... I commented that they were irrelevant to, and contradictory ... existed for over 2 millennia without needing any infinite naturals. ... A natural number is finite if the set of naturals less than or equal to ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... The same axioms that give us transfinite set theory are needed to ... give us the naturals, and from them the other number systems on ... The invertible functions of IFR map reals to reals. ... infinite number. ...
    (sci.math)
  • Re: How seriously do you take mathematics?
    ... >>> naturals as naturals, everywhere on the number line, that there are equal ... >>> Is it possible, as set theorists seem to believe, that an infinite ... > In an infinite binary tree, as you asserted, there ARE no terminal nodes, ... >>> When you look at this set of axioms from a purely logical ...
    (sci.math)
  • Re: infinitely many nns = infinite nns?
    ... than the naturals in any case. ... infinite sequences OF RATIONALS ... quantifier and an additional quantifier on a property ... MORE than just these 5 axioms. ...
    (sci.logic)
  • Re: Enumerable sets
    ... The infinite ones have the SMALLEST ... naturals themselves, or the set of all strings over an alphabet, ... set of axioms by use, say, of a finite number of axiom schemata, ... I specify a denumerable set of axioms that is not recursively ...
    (sci.logic)