Re: Calculus XOR Probability

imaginatorium@xxxxxxxxxxxxx said:

stephen@xxxxxxxxxx wrote:
Matt Gutting <tchrmatt@xxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Matt Gutting <tchrmatt@xxxxxxxxx> wrote:
Tony Orlow wrote:
MoeBlee said:
Tony Orlow wrote:
MoeBlee said:
Tony Orlow wrote:
To say
all elements are finite but the set is infinite implies that there is some
lartgest finite with an infinite successor.
In what theory? In set theory? No, what you said is not true of set
theory. So in some other theory? If in some other theory, then what are
its axioms and primitives and what are its definitions of 'infinite',
'largest', 'finite', and 'successor'?
I was speaking of the logic behind the limit ordinals, and how it is flawed.
You said "To say all elements are finite but the set is infinite
implies that there is some lartgest finite with an infinite successor."
That statement contradicts set theory. So my question, which you did
not answer, is: In what theory do you cliam that your statement holds?

And there is no flaw iin the logic behind limit ordinals. The only
logic involved in set theory is first order predicate logic.
I suppose what it boils down to here is infinite induction. Since the size of
the set is the successor to the maximal element in all finite sets, this
relationship should hold, being an equality, in the infinite case. If that is
so, then omega is successor to the largest finite, and the notion is self-
contradictory. Infinite induction appears to be discredited, but the only
counterexample in this thread is easily explained otherwise, and "infinity did
it" doesn't fly when it comes to explaining an error of sqrt(2). The limit
ordinal omega is in direct contradiction with infinite induction, so one of
them is wrong. Hint: it's not infinite induction.

The problem I see with this is that the finite cases, as you yourself
point out, deal with sets having maximal elements. Since this is not true
for "the infinite case", I don't see how you can apply the same reasoning
there.

Matt

But Tony's infinite sets do have maximal elements, such as
{1, 2, ... BigUn }. Tony is absolutely incapable of imagining a set
without a maximal element. Everything must have two endpoints and be on
"the number line" or else he is simply incapable of understanding it.

Basically Tony like to use the word infinity, but he really does
does not understand or believe in the concept.

Stephen

But apparently even Big'Un isn't a maximum element, since you can go
beyond it by adding another unit infinity. It looks like the system has
"endpoints" which are functionally not endpoints (since you can surpass
them) but which are nevertheless declared to be so.

Matt

Big'Un is not the maximum element in an absolute sense,
but when you "declare" an infinite set, you have to specify
a range. Big'Un is apparently the default.
The point is that in Tony's system
{1, 2, 3, ..... }
is apparently meaningless. You have to specify some upper bound
such as Big'Un, or Big'Un+1, or whatever before you can say
anything meaningful about the set. A set must have a maximal
element. That is one of the absolute rules of Tony's logic,
and one of the main reasons he rejects standard set [theory] is because
it allows for sets that do not have a maximal element.

Hmm, I'm not sure the rule is that the maximal element must actually
exist. There was certainly a stage at which Tony would agree that it
didn't exist. But the rule appears to be that - existent or not - you
are able to refer to it in the course of an argument.

Brian Chandler
http://imaginatorium.org



Boys, the rule is simple. If you are going to measure a set by a mapping
function from a standard set, that mapping function gives a measure of the set
density at any given point. In order to measure the set using the density of
one sort or another, one must have a value range. It's not that sets with no
defined reange can't exist. One just cannot acccurately measure them, except
using the range as an indpendent variable and applying the mapping over that
range. Big'un serves as a declared unit infinity, used for measuring and
comparing infinities, declared and compared formulaically.

--
Smiles,

Tony
.

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