Re: Logarithm of transfinite numbers

Tony Orlow wrote:
Randy Poe said:

Tony Orlow wrote:
Randy Poe said:

All we know from your first statement is that no terminating
sum can represent an infinite value. It does not follow that
the sum over all finite bits is finite.

Forget temrinating sums.

Every sum of the form sum(k=1..,M) a_n is a terminating
sum. Why is it I should forget them if it is you who are
attempting to draw a conclusion from them?

You are hung up on the notion that you can prove
anything you want based on the lack of a lrgest finite.

No, I'm "hung up on the notion" that there is no principle
that says something true of a certain collection of proper
subsets has to be true of the set as a whole. I could
take a subset of humans, see that they're all blue
eyed, and try to conclude that all humans are blue-eyed.
But no logical principle lets me do that.

If you can determine that there are no humans in the set with eyes that are not
blue, then you can say that blue eyes is a property of the set of all humans.


Grow up. If there is no
finite natural at which point the set is infinite,

Another way of saying that the set N is not one of the
set of finite subsets {1,...,n}

No, that is an entirely different statement, and a leap without justification,
or even rationale, on your part.


and finite naturals are the
only points in your set, the set can't become infinite at any point in the set.

No, the sequence of subsets {1,...,n} can't become N.

N is the limit of those sets as n approaches aleph_0.


But that doesn't mean N doesn't exist. Nor does it say
anything one way or the other about N, a set which is not
in this sequence.

It's not a sequence of initial subsets. I am getting sick of your circular
thought pattern and refusing to acknowledge that this is a property of the
finite naturals that impacts the size of the set. Forget the minute technical
difference between a terminated initial subset and the entire set. The
terminations of the subsets are the naturals, and the property pertains to
them.



So, no countable string can represent anything but a finite value. Exactly
which statement is wrong?

The one that says "it requires an infinite position to
achieve that". Also the one that says I agreed with that
absurd claim.

Then explain how something occurs in a set of finite bit positions which cannot
occur at a finite bit position. Stop dancing around the question.

I'm not "dancing around" anything.

You are. You're hanging onto this objection regrding the difference between N
and any initial subset, when that's entirely irrelevant.

The central point I make
over and over is the same.

Tell me about it.

There is no reason to expect
that something true of the set as a whole "occurs at"
an element, or a proper subset of that set. The whole
set is a fundamentally different thing, as different as
cats and dogs, from the particular subsets you keep
focusing on. You WANT there to be a principle which says
"if it's true of the set, it must be true of one of this
particular class of subsets". But there is no such principle.

If a property of the set relates to the set size up to that point, and says
something is true for all elements of the set, then that fact about the set
size at every single point in the set includes EVERY point in the set, and
pertains to the set as a whole.

They are no more related than cats and dogs. You
are hung up on repeating this principle over and over,
but

it

does

not

exist.


It does. All you have in the set are finite naturals, and an infinite set size
does not occur at any one of them, so it does not occur in the set. Sorry,
Charlie.
- Randy
Tony

Well, I'm a bit confused by these three characters. Who's Charlie,
Tony?

Anyway, please tell me a little more about the property of Orlovian
"unboundedness". I gather you make some claim that in your "system",
the set of finite naturals is "unbounded". Is that right?

Can you then give me an example of a finite natural, in the set of
finite naturals, that is "unbounded"? I'm rather guessing that since
it's possible that we all mean the same thing by a "finite natural",
that is, something which can be represented by a string of (decimal)
digits, with two ends, and such that given enough time one could read
from one end to the other and stop, you will say that there are not any
specificiable, actual, finite naturals that have this property of
"unboundedness". Is that right?

In particular, why does the following argument not show that the set of
finite naturals cannot be unbounded:

For every finite natural x in this set, it is true that the set of
finite naturals up to x is bounded, since after all, none of them is
more than x. This property relates to the set up to this point, and
what's more this is true for any value of x. If a property of the set
relates to the set size up to a point, and says something is true for
all elements of the set, then that fact about the set size at every
single point in the set includes EVERY point in the set, and pertains
to the set as a whole.
Therefore the set as a whole cannot be "unbounded" in the Orlovian
sense.

Brian Chandler
http://imaginatorium.org

.

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