Re: An infinite set is always "maintained"

On Sunday, September 25, 2022 at 1:40:57 PM UTC-7, Ross A. Finlayson wrote:

On Sunday, September 25, 2022 at 12:19:13 PM UTC-7, Sergi o wrote:

On 9/25/2022 1:26 PM, Ross A. Finlayson wrote:

On Sunday, September 25, 2022 at 10:21:03 AM UTC-7, Sergi o wrote:

On 9/25/2022 11:29 AM, Ross A. Finlayson wrote:

On Saturday, September 24, 2022 at 1:08:18 PM UTC-7, Sergi o wrote:

On 9/24/2022 2:44 PM, Ross A. Finlayson wrote:

On Saturday, September 24, 2022 at 12:41:27 PM UTC-7, Sergi o wrote:

On 9/24/2022 2:30 PM, Ross A. Finlayson wrote:

On Saturday, September 24, 2022 at 12:20:13 PM UTC-7, Sergi o wrote:

On 9/24/2022 2:08 PM, Ross A. Finlayson wrote:

On Saturday, September 24, 2022 at 11:20:15 AM UTC-7, William wrote:

On Saturday, September 24, 2022 at 10:20:13 AM UTC-3, WM wrote:

William schrieb am Freitag, 23. September 2022 um 21:17:13 UTC+2:

On Friday, September 23, 2022 at 3:47:51 PM UTC-3, WM wrote:

William schrieb am Freitag, 23. September 2022 um 18:13:23 UTC+2:

On Friday, September 23, 2022 at 9:40:12 AM UTC-3, WM wrote:

If there is no last element, then there is no intersection,

Nope. The definition of the intersection set does not require there be a last element

The definition of inclusion-monotonic sequence requires that alle elements which do not disappear somewhere are in all terms.

Since there is no element that does not "disappear somewhere" there is no element which is in every term.

If every element disappears somewhere,

Which is does

then the remainder is empty.

Correct (note that the remainder is not a set of the sequence

Then the intersection is empty too.

Correct.

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Seems you never heard that direct addition of sets was a natural operation,

actually it is called Union, a common set operation, and I'm sure he has heard of it.

and some have infinitely-many copies infinite, others, empty, ..., but let's
leave that out of whether or not there's the "actual", infinite.
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Just pointing out that there's a formalism over your transitive,
results opposite your inductive inference with deductive inference.

no inferences, just facts. I find if I correct your above sentence, there is nothing left, it is incorrectable.
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an unfortunate grouping of babblated words occupying disparate servers disk space.

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No-infinite-descending-chain is independent your inference, ....
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Also there's lots of higher mathematics and plenty of it that
reverses X's and O's and finds the same dumb argument.
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Seems you never heard of reverse. (And don't have an infinity.)
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Or, excuse me, that's too stupid for what all infinity includes.
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There's a bigger, better mathematical world than you just left the room with.
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And it's not much more what you just bought.
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Where I am located does not effect the mathematical world, unless I am giving a lecture.

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And then you insult me?

not at all, why should I do that ? you have risen above the Xs and Os, and provide a higher level, a different view. A try to widen perspective.
with WM, it is basic math he flounders upon, so the discussion is on basic math. and unfortunatly, with WM threads, it will never get to higher math
(algebra!). I try to widen it out at times, to fields, vector calculus topics, but that horse dont walk that way

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Oh, when you say "that's gibberish" it's an insult.

But I did not say that's gibberish, I said something completely different and positive;
"an unfortunate grouping of babblated words occupying disparate servers disk space."
If it is on a lot of servers, it must be important to someone.

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If not to me to somebody who reads it and "I didn't think it was gibberish, ...".

anybody ? post here to reply.....

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It's usually just that making theorems for open over closed topologies
is deciding all one way, but, each entire structure has its own all one way.

do you mean like a Diode one way, or a one way Street ?

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What establish neighborhoods according to delta-epsilonics what
makes "calculus".

which neighborhoods are those ?

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So, ..., mathematics has _both_ line reals and field reals.
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Now you might say "that doesn't make more mathematics, it only
reduces what was the mathematics down to even more derived",
"that both decides and makes independent usual conjecture there's
an infinity", indeed I have made a smaller more personal mathematics,
then all classical and including all the rest of the course already, also.
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Instead of adding it after, there's the "rest of the foundation", before.

no wonder!

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So, I don't "claim mathematics" any more than having one, but, have all
the requirements of what any mathematics must be, so includes this all.
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That is, to be a _foundation_ and not just another hedge, all these properties
"before and after set theory", must be a geometry.
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Then it's like "damnit Ross you can't have taken all the theory" and yes I did.
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Damnit Ross, I took all the theory, and put it into a simple one, for a small head.
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It's called the "slates" for uncountability and what was logical paradox, that
condense why line reals are consistent again, and, make for independent,
naturally, all objects of free _thought_. ["Expansion of comprehension."]
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It's mostly according the principle of "sweep": course-of-passage in "all the
ordinals" natural in time. That's only because ordering theory and set theory
exchange order in type.
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Then before regular cardinals meaning "my first transfinite cardinal an object
in set theory", not "regular, ... but large cardinals", there is that _EXACTLY
ACCORDING TO WHAT WOULD BE RUSSELL's PARADOX_, the "finites" is
already: "infinite". (Also not regular.)
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It "results" then that that "is" its real property.
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There's for example "ah ha no standard model of integers, all larger or smaller",
"as if you didn't know, a standard model of integers is imperfect or finite".
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It would be best to put items that may be considered by others as gibberish, in Quotes.
It would clarify your verbatage to the inexperienced reader of math fiction.

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So, I have tons of requirements what a mathematical foundation is,
when you beat a simpleton and don't instead gently provide a definition
that results explaining the invalidity and upholding the validity both,
of the view, what results is "if you don't know X's and O's how can you
know p's and q's"?
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The X's and O's are _reversible_ it's a usual principle exists _inverse_.
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Where they're not those are X's and "no".
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Because, as a matter of fact, for some people or anybody, making axioms
like these reading off properties oe quantification, introduces a whole
new foundation next to set theory called bigger set theory.
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Which results _less_, also better.
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I don't know how you have sequences with no ends, but here when there are
two ends, then the sequences go in the middle, and, ..., exists a middle.
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Exists a middle.
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Now write the rest, ....
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This way I came to understand mathematics and poster-childed myself for the 20'th C.
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you can be of great help to WM, keep him on the right path.

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And solve and resolve the paradoxes of mathematical logic?
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And show that "complete ordered field + least-upper-bound + measure 1.0"
isn't the only way to arrive at a continuous domain and measure 1.0?
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And show that together it must be this "extra-ordinary theory" first?
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There's a great debate in mathematical foundations, and I already had it.
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Now, when I say, "it's a theorem there's no standard model of integers",
it actually is. That there's a model it's extraordinary.
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I do _not_ write gibberish, and the more you read you'll find I had the _last_ word.

I am glad you think you do not write gibberish.
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gibberish
jĭb′ər-ĭsh
noun
related to gibber
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1 Unintelligible or nonsensical talk or writing.
2 Highly technical or esoteric language.
3 Unnecessarily pretentious or vague language.
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which category do you think it fits in ?

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And sometimes the first, ....
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So if you're interested in true mathematical foundations,
that's great and I'd encourage it, but if it isn't all true, mathematical,
and foundation, all the foundation, it's not.

poured, or poor foundations ?

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When all it can be is proper axiomless natural deduction, ...,
the primary theories sort themselves right out.
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So, pick your nose at that.

I have people that do that.

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Primary theories here include set and part theory, geometry, type theories
and categories,

type fonts ?

number theory, order theory, operator theory, function theory,
topology, the list doesn't go on. Most opinions are set in topology for function theory.

no. read "the theory of functions" by knopp first, and you will change your mind

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All one theory, ..., best.
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How about "The Theory of Real Functions"?
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How about read the Wiki on infinitesimals,
and see now that 1/infinity is since the 1600's
and that the infinity symbol or lemniscate is
same.
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The Wiki used to say "infinitesimals might be zero",
now the tide pulls "infinitesimals are non-zero",
either one way being wrong, where might
be _not_ and are _not_.
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Since 1600's now there's Wallis, since MacLaurin too,
infinity's the same old symbol; and has its direct meaning.
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Here function theory is split "DesCartes" and "not DesCartes".
DesCartes is "provide a space of all the functions".
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There's a split in function theory in ZFC its theory of functions.
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So, the Wiki at least is all "..., historical".
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Tower of Rain - , ....
House of Cards / Tower of Rain
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Alright, this Carus, "A Primer of Real Functions", Boas, MAA, 1981,
have at it.
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What category your pretentious gibberish fits in is who cares.
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Don't you have an entire theory of understanding it?

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You know what it used to be when you looked that up?  "Go to hell."
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You'd research the historical argument and get "WRONG".
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Now it's "KNEW ALL ALONG".
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There's your progress, ....
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You know why Cantor banned infinitesimals from his theory?  Because it broke.
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He was perfectly happy to try to build to them all constructible,
elements of analysis of elements of integration, what they are.
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In their fields, ....
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Then it takes at least having "the most effective infinitesimals"
that are individua of a continuum that sum to it.
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Cantor was perfectly happy to have no universes in his theory,
be incomplete, ..., and uncountability as "totally fundamental".
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Being internally self-consistent, though, isn't same as being
complete and whole, the theory of all mathematics, or "the
set that is set theory".  Also the same all mathematical paradoxes
why there is class-set distinction and so on, why CH is un-decideable
and so on, they are just sitting there.
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Still, it's a very strong theory, and with the support of Zermelo and
Fraenkel, there is ZF set theory with Choice also being required usually,
which mostly just means ZF is strong enough to be a theory of ordinals.
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An overall curriculum was carefully cultivated from it, and it resulted
a lot of strong and weak results, then that in a convention, the
continuity the mathematical property, is firm enough according
to density, that countable additivity results integral sums.
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There's an easier first one though, continuity first.
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