Re: infinity

>> I suppose I can see that, in theory, but it seems to me that in many
>> ways the infinite reals and naturals may be treated the same as the
>> finites, and that these systems shange treatment arbitrarily in ways
>> that lead to nnsensical conclusions. Wouldn't it be better if we had
>> one consistent way of dealing with infinite quantities? I think so.

Perhaps, but how is one way better than another. Non-Standard Analysis
is perhaps the most "natural" in an arithmetic sense, in that it
perserves most of the known laws of arithmetic. On the other hand, the
most natural from a "size" perspective is cardinality and are (to me)
the most "real" of the infinite systems, as it is the one that is most
widely used across different areas of mathematics. The ordinals are an
interesting phenomenon, but to me that are useful only in as far as a
bridge to get to cardinal numbers. The Extended Real line adds the
ideal point at infinity to give the real line topological equivalence
to the circle (likewise the Extended Complex plane becomes
topologically equivalent to the sphere), and has almost no noticeable
impact upon arithmetic. I don't think there is a need to choose.
After all, since there are many finite numbers, why should there be
only one infinite number?

>> What part of my reasoning is inconsistent, specifically, in your
>> opinion?

First let me point out that I think you can consistently create a
system using some of your ideas, but most of the inconsistencies come
(I think) due your adherence to past statements which cannot withstand
critical examination. For example, you state that the the integers are
"finite but unbounded", which is clearly a contradiction in terms. Now
you may wish to ue your own definition of finite, say TO-finite, which
means something other than the standard definition (Dedekind did the
same kind of thing). But there is already a definition of "finite" and
"infinite", and they are not yours. Pulling stunts like that will
immediately place you into the halls of crankdom.

Secondly, instead of following mathematical procedure (start with
knowns and prove conclusions based on these), you are playing politics
and religion (start with your pre-determined conclusions, and then look
for any arguments which support them, consistent or not). This is not
how mathematicians operate, nor will any take you seriously when you do
this. You postulate a system where there is a largest integer and no
smallest infinity (the opposite of what we have in ZFC), and the
natural question is: how did you arrive at this? If the only answer
is: "I don't like the conclusions of the other system", you are
engaging in religion, not mathematics. Frankly, there is nothing wrong
with postulating a system where you end up with a largest integer and
no smallest infinity, and it certainly will have different properties
than the current system. But unless you are rigorously following the
rule of logic to get there, all you have is essentially the
mathematical equivalent to a fiction novel. Honestly, what reason do
you think a mathematician would have to consider what you're saying, if
you are playing so fast and loose with the truth?

I do believe you could architect a consistent system, using different
axioms and have surprising conclusions, but what you have so far is (I
am sorry to say) nothing more than a footnote in newsgroup crankdom. I
believe you have the intellectual capabilities to break of of this, but
I am also aware of the psychology behind so desperately clinging to
this world view, that it may be beyond your emotional capability to
give up. I don't know, that is a decision for you to decide.

People familiar with sci.math know of many cranks here: James Harris,
Archimedes Plutonium, Nathan the Great, Ross Finlayson and others.
They will take up a lot of space on this newsgroup, but they will never
be taken seriously as mathematicians. They won't, simply because they
are unwilling (or unable) follow mathematical and logical procedure.

Hope that helps (yeah, I know, I should find a new tag line),

Jonathan Hoyle

.

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