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File: realsarescam.png (44 KB, 1200x1193)
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explain to me what a "real" number is without using any schizo infinities

>pro tip
u cant
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>>12573322
Another hoax made by scientists who crave money and recognition over truth, what a surprise.
>>
>>12573322
The reals are a inconsistent system that is a
delusion to pretend to understand something we don't.
>>
In truth science is not equal to reality, since reality is beyond the imagination of scientists.

Hence, no real numbers exist, just like any "real" theory. You believe a theory to be about accurate or you do not. Common sense makes you choose.
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>>12573349
Amateur assumes you can believe in the first place. Get back to wagecucking slave.
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>>12573322
just a point on the line
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>>12573376
Yeah, you are being very smart now. Stop being angry, such emotions make you stupid.
>>
the real numbers are a complete ordered field
a real number is an element of the real numbers
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>>12573383
Is the red point rational, as a real number?
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>>12573389
>complete ordered field
And what is a complete ordered field?
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>>12573392
thats not a point. thats a neighbourhood of a point
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>>12573397
Ok then is the point of which that is a neighborhood rational or not?
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>>12573396
look it up its on the internet retard
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>>12573400
could be
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>>12573401
Wikipedia says a complete ordered field is a certain kind of field which is a certain kind of set. And what is a set? Wikipedia doesn't give a proper answer.
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>>12573406
a set S is a symbol charaterised by the symbols x for that the expression x [math] \in [/math] S is true
>>
The closure of the rationals
>>
It's the sequence. Its not the number. It's the algorithm.

Wildberger is right about them. But he is wrong about rationals, they too are vague non-numbers. The problem of the real numbers are the rational numbers.

Only numbers are: none, its successor one, and their successors. Nothing else.


btw. there are no set of natural numbers. Sets are finite.
>>
>>12573420
The closure of the rationals is the set of rationals.
>>12573419
What kind of symbols are allowed? What does it mean for an expression like " x ∈ S" to be true?
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>>12573430
>What kind of symbols are allowed?
No restriction

>What does it mean for an expression like " x ∈ S" to be true?
It doesnt "mean" anything. Its just something that is assigned to the symbol S
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>>12573430
tell me what is the solution of x^2-2 =0 the rationals

if they algebraically closed this should be easy
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>>12573428
>But he is wrong about rationals, they too are vague non-numbers
What's wrong with saying a rational number is a pair of integers p,q, q not zero, and a calculation that shows p,q have no common divisors (can be made more precise by saying a list of pairs of integers (p_i, q_i) starting from (p,q) where at each step you subtract one from the other, and end up with a pair with 1 in it?
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>>12573441
>No restriction
Ok so what sets are ₧, , , ?
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>>12573430
By closure, I mean the set and it's limit points. For any real number, we can create a sequence of numbers that converges to it with error ε small (by the Archimedean Property).
So if we consider the set of all convergent sequences of the rational numbers, L, then the reals are the union of L and Q
>>
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>>12573322
if you agree with figure 1 the you must agree with figure 2
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>>12573442
>x^2-2 =0
It doesn't have a solution in the rationals. That's one of the most popular proofs of all time, due to Euclid. Surprised you aren't aware of it.
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>>12573448
>Ok so what sets are ₧, , , ?
it depends on the context
here they have not been defined so i cant tell you
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>>12573457
>By closure, I mean the set and it's limit points
Yes, the set of its limit points is the set of rational numbers. You haven't added anything new.
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>>12573465
But you said sets ARE symbols. What do you mean I haven't defined them? According to you, I have you a set. A set that IS the symbol. Now tell me what kind of set it is. Is it an infinite set?
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>>12573462
but you said the rationals are their own closure. curious!
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>>12573475
you have not characterised it yet so its not a set but just a symbol
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>>12573446
how can ONE number be a PAIR. It's just illogical.

Only define natural division.
A is a Nat. B is a Nat>0.

Then A/B=C where B*C is the highest possible less than A and C is a Nat.

Also A%B=D where D is a Nat and the leftover of previous.
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>>12573479
Yes, any topological space X is its own closure, since the whole space is always closed (because the empty set is open in the definition of a topology). Closure is only interesting with smaller subsets.
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>>12573482
What type of characterizations are allowed then? Can my characterization be "₧ is the set of all sets that don't contain themselves"?
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>>12573487
interesting
whats the basis of this topology
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>>12573497
Doesn't matter, in any topology you want to give it, the closure is going to be the whole space.
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>>12573458
You think you are clever? Both are results of geometrical constructions, results are sequences and the other just happens to have in this number system.

Also famous NUMBER THEORY problems considers only natural>0 solutions
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>>12573458
You look at quadratures instead. "length" isn't a well-defined concept.
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This all mathematical fantasy garbage of real numbers breaks in computations. The error is meaningful and has to be carried and avalanced till the very end and then represented. There is no way one can say that epsilon is zero. Machine epsilon definetly is not.
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>>12573322
point on a number line
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>>12573537
Where can I find this number line? How do I know it exists?
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>>12573519
How did he even proof quadratures?
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>>12573322
What no pussy does to an mf
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>>12573543
you must be over 18 to post here
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>>12573322
The real numbers are the numbers that are real.
I'm an EE so werks 4 me
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>>12573494
only the ones in ZFC are allowed
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>>12573670
>only the ones in ZFC
explain what you mean
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>>12573677
the characterisation of a set can only be done according to the rules that the axioms of ZF(C) establish
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>>12573686
So there are only a countable number of sets?
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>>12573702
fuck I lost this time but Ill be back with a more autistic theory
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>>12573322
Everyone in this thread is a delusional faggot. Real numbers are the set of all possible limits of Cauchy sequences of rational numbers. If that is not suitable for your autistic brains, then google Dedekind cut and dont bump a thread with more value that this shitpost off the board.
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>>12573770
>possible limits of Cauchy sequences
And what is a sequence?
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>>12573770
Everyone ITT already knows about these excuses of a definition such as "equivalence classes of Cauchy sequences" and "Dedekind cuts". These are fake and don't work.
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>>12573808
>These are fake and don't work.
nice argument
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>>12573809
Try and properly explain any of them. Protip: you can't.
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>>12573809
The notion of a sequence is meaningless. It does not have a definition.
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>>12573798
a countable set of rational numbers
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>>12573322
a real number is 2+2=4

and an irrational number is something like 2+2=5
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>>12573543
Draw with pencil. But is just a line. Not a number
>>
Best definition of [math]\mathbb{R}[/math] here:
Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis
https://vixra.org/abs/1906.0237
I define them as the set of possible limits of a certain Cartesian product of Cauchy sequences in union with the set of all modified Dedekind cuts. It's pretty much like: >>12573770
but I fix a big problem in the direct Cauchy sequence/Dedekind cut picture. The result is A product of sequences and a minimally modified cut.
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>>12574209
And what is a Cauchy sequence?
>>
>>12573322
Obviously is impossible to have a finite realization of a infinitary concept. The problem is not the infinitary concept, is wanting the finite realization, just like wanting an eternal being to give up its inmortality and have a human time scale existence.
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>>12573798
>>12573798
it's like jazz, if you gotta ask you'll never know
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>>12573941
based
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>>12574220
I think it's pretty lame how people grant that there is a common knowledge base with which they are already familiar such as the set of characters in the Latin alphabet and words in the English language but then as soon as you start talking about math, they act like you're some preposterous asshole to assume a shared knowledge base of primitive concepts.

I wrote this paper one time where I said, "A certain surface in a bulk is a delta-function." Everyone was like, "Oh, that's wrong and stupid," or, "vague and meaningless," maybe, like they don't have any preexisting knowledge of delta-functions by which they could understand how a surface in a bulk "is given by" a delta function because I only wrote "is." For instance, you can pull the volume of all of 3-space at time t_0 out of an integral over all of spacetime as in pic rel where the delta function specifies the hypersurface of constant proper time at t_0. People acting like they don't know the relationship between a delta-function and a surface in a bulk are just being deliberate retards like you saying, "I don't know what a Cauchy sequence is because you didn't explain it." I didn't define the characters in the Latin alphabet either but somehow you're able to read what I'm writing. How is that, do you think? Overall, I'd say arguments of the form, "I don't know what a Cauchy sequence is," or, "I'm not aware of the one-to-one correspondence between a surface in a bulk and a delta function," are designed to prey on the ignorance of people who don't know that these are kindergarten-level concepts in the shared knowledge of the intended audience... as are the Latin characters and English words with respect to which no one ever says, "I can't read what you wrote because you didn't define your alphabet," or, "I can't read what you wrote because your brief description of the alphabet as a segue into the point of what you were writing was insufficiently rigorous."
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>>12574301
This is literally the *only* thing that delta-functions do. You're a professional scientist and you can't understand the sense in which a surface is a delta function?
>>
And certainly it is true that a surface is not a delta function but when they fixate on my having written "is" rather than "is given by" in a barely relevant point raised in the introductory paragraph, it's like someone writes a PhD thesis and their advisor throws it in the garbage because they left the phrase "Cauchy sequence" undefined.
>You said {x_n} is a Cauchy sequence
>that's wrong
>Cauchy sequences aren't defined.
>into the trash
It's even worse than that because this was not a math paper and in physics it is normal to abuse not only the language, but even the math itself.
>>
ITT: burger gets btfo by tooker
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>>12573322
Explain to me what a natural number is without assuming they already exist.
>pro tip
u cant
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>>12574429
A string of strokes.
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>>12574429
A pile of rocks.
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>>12573392
no, the probability it is rational is 0
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>>12573404
in reality, no
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>>12574432
How do you tell apart two strings of strokes?
>>12574450
How do you tell apart two piles of rocks?
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>>12574479
>How do you tell apart two piles of rocks?
Pick a rock from one pile, put it aside. Pick a rock from the other pile, put it aside. If one of the piles is empty and the other is not, the piles represent different numbers. If neither is empty, repeat from the start.
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>>12574493
>Pick a rock
How?
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>>12574497
In the pile there are rocks. You just take one of them.
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>>12573392
>>12574468
The answer is that its rationality (or lack thereof) depends on the chart.
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>>12574534
>one
So you're using natural numbers to define natural numbers?
Righto, schizo.
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>>12574542
I'm not. I'm talking about an object, a rock. Taking a rock from a pile. You know what that means.
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>>12574546
You didn't define it, so no I do not. You are the one claiming that math needs more rigor, yet you do not provide any more rigor than ZF(C).
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>>12573406
an ordered field is a field that holds the property that if x and y are positive numbers, then xy is also a positive number. It also means that if you have numbers x,y,z where y < z, then x + y is less than x + z. It just adds additional structure to a field which does not necessarily have any notion of order
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>>12573322
tl;dr if it stops its real
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>>12574559
Define what? A rock? Do you not know what a rock is? How about a stone?
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>>12574623
You didn't define "one". You also didn't define "take a rock from a pile". How do I know that I can take a rock from a pile? How do I distinguish between taking one rock and taking multiple rocks?
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>>12574638
Define define
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>>12574754
Ask the same "Define define" but without using any words or codes
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>>12573826
Let S be a set, and let N be the set of natural numbers. Let f be a function with domain N and codomain S. Then f is a sequence.

See?
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>>12573322
>what a "real" number
Only relevant b/c of squareroots and exponents/logarithms, which are undefined until you get to the extended complex plane
i.e. a middleman set for midwits
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>>12573322
>b-but muh infinity
Take your meds
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>>12573322
a rational or irrational number
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>>12573322
a complex number where the imaginary part is zero
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>>12575208
>irrational number
Surely you didn’t fall for that meme. How can a number that can’t be expressed as a ratio exist?
>>12575212
>complex number
>"imaginary" part
Do you even consider how absurd this sounds? How can I multiply two numbers together and get a negative one? Please cease these schizophrenic ramblings.
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>>12574209
>vixra
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>>12575256
>>12575200
>>12574789
>>12574754
>>12574638
>>12574623
>>12574559
>>12574559
>>12574538
>>12574534
>>12574497
>>12574493
>>12574479
>>12574473
>>12574450
>>12574432
>>12574429


stupid chuds, just assume the natural numbers exist and 1+1=2, there is really no room for debate on those fronts, you either believe it or you dont.
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>>12575340
Right. And it's no less reasonable to just assume the axioms of ZF(C).
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>>12573322
>dooot doot look at me, I don't believe in infinity, I'm a faggot, la di da
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>>12573322
imagine getting into ultrafinitist crankery without even a high schooler's understanding of mathematics.
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>>12573322
0.999 != 1
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>>12575414
>doot doot look at me, im schizophrenic and believe in delusions
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>>12573322
the non imaginary part of a complex number
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>>12574497
buridan's ass just got stupider
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>>12575350
Prove it.
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>>12575857
Well, I assume that it's true, and it follows immediately. Just like what you do with natural numbers.
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>>12576046
That's wonderful. You can just wish anything into existence. Here's a proof that twin prime conjecture is true: I just assume it's true! Booza, now it's true, and the proof is no less reasonable than simply assuming the axioms of ZF(C). How do you like them apples?
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>>12576067
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>>12575334
One of these days, your relatives are going regret that you wrote such things.
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>>12576067
It is much less reasonable because one has intuitive (natural) reason to believe the ZF axioms are true. Axioms supposed to be minimally assumptive but your axiom that TPC is true is highly assumptive.
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>>12576155
>It is much less reasonable because one has intuitive (natural) reason to believe the ZF axioms are true
Go on, explain the natural reasons to believe the ZF axioms are true.
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>>12576302
You can read them and say, "Ok, that makes sense."
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>>12576735
The axiom of infinity doesn't make sense to me. Explain what about it is intuitive to you.
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>>12576738
All non-infinitely ordinal sets have a natural number of elements. If there was a greatest number of elements in a set then that would be a special natural number. There are no special natural numbers. The number of possible elements in a set is unbounded. The greatest number of possible elements in a set is infinity. Duh.
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>>12576751
By a set I imagine a collection of objects, like a collection of keys or whatever. So yeah, there is no bounds to how large a set can be. But from this it does not follow the existence of an infinite set, whatever that means.
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>>12576757
"Unbounded" and "infinite" are synonyms. Aside from that, consider the set of natural numbers [math]S=\mathbb{N}[/math]. If you don't believe in infinite sets, then you must believe there is a highest natural number (unless you have some weird workaround in your thinking.) The idea of a highest natural number is unintuitive to me.
>>
>>12576766
I believe in natural numbers, but I don't see how it makes sense to complete them all into one object, a "set of natural numbers". What is the justification to doing this, especially when we know asking a lot of even the simplest questions about this object are still considered open problems today? Clearly there are no actually infinite collections of anything that we have access to and can manipulate, so we must give up the link from mathematics to reality. But then how do we know that the theory is even consistent?
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>>12576776
Lol, what do you call the symbol [math]\mathbb{N}[/math]? Since math is abstract, what about the set of all points in a line segment? What would you call the collection of all those points if not a set?

>there are no actually infinite collections of anything that we have access to and can manipulate
I disagree. What about the set of all positions between two marks on a ruler? In quantum mechanics the state of being at each of those positions forms a Hilbert space such the set of the Hilbert space's spanning basis vectors is an infinite set. We can definitely access and manipulate these states. They've been doing it for about 100 years.

>how do we know that the theory is even consistent?
You can do direct experiments to show that position eigenstates are accurately treated as orthogonal directions in a infinite dimensional Hilbert space. The basis vectors of any vector space can be assembled into a set. The set of the basis vectors in the state of positions between two marks on a ruler is an infinite set. To show that this is not some one-off coincidence, you can do the same thing with the spanning basis vectors of an infinite dimensional Hilbert space of momentum eigenstates. A Hilbert space is just a vector space and it is totally Cartesian in nature so we're not doing anything tricky between real analysis and quantum mechanics: we can easily test the validity of theories which use the idea of an infinite set.

How about the set of all energy eigenstates for an electron in a hydrogen atom? He we can verify the theory again for the case of a countably infinite set of basis vector in the Hilbert space as opposed to the above two examples regarding uncountably infinite sets of position and momentum eigenstates.

Fundamentally, not being able to disprove the theory doesn't prove that there are infinite sets. The thing that "proves" the existence of infinite sets, for the purposes of allowing an intuitive axiom, is that "unbounded" means "infinite."
>>
Basically, if you're thinking of sets as collections of physical objects like keys then you're always going to bump into the size of the universe as a limit on psychological abstraction. In fact, psychological abstraction is not bounded by the size of the universe and math is a tool for psychological abstraction so it's better to think of sets of abstract objects: the set of natural numbers is the main example here. I think it's silly to say, "The collection of all the natural numbers is something other than a set," or, "[math]\mathbb{N}[/math] is ill-defined." If you're thinking about collections of keys, long before you get to the size of the universe, the gravity of the keys is going to melt them together and change the number of keys.
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>>12576776
Mathematics is the realm of abstract ideas. As long as the system is consistent, it should not matter whether or not we can find "infinity" in reality. From the mathematicians perspective, it is an interesting idea to explore. The properties of a mathematical object are not necessarily shared by anything that "exists" in external reality.

For other disciplines, mathematics is used as a tool. It may seem strange to describe the physical world using these ideas, but ultimately it is only a model. The model is not meant to be an exact and completely accurate description of reality, no more than a map is supposed to be a perfect copy of the terrain it describes. It is the work of the scientist to determine which models work best and for which purposes. If it allows us to make predictions and those predictions seem to align with what we observe, then it certainly has utility, even if there is no such thing as a true, completed infinity.
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>>12573322
the reals are what the rationals are not.
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>>12576933
Stop being a retard who confuses poetry with logical thought.
Rationals are NOT numbers such that a+b=/=b+a. Are you saying that the Reals ARE such numbers?
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>>12576878
>Mathematics is the realm of abstract ideas.
Agreed. Nothing wrong with abstract ideas, mathematics couldn't do without them. I'm actually a Platonist, I admit the existence of abstract natural numbers independent of their implementation.
>As long as the system is consistent, it should not matter whether or not we can find "infinity" in reality.
Ok and we know that the system is consistent how exactly? (we don't)
>For other disciplines, mathematics is used as a tool
Yes. Nothing wrong with mathematics as a tool. People who actually use maths for practical purposes don't give a shit about axioms or set theory or whatever. The problem is with pure mathematics, claiming something is rigorous when it's not. Mathematics, as it currently stands, is without a foundation.
>but ultimately it is only a model.
This is just ridiculous. What does set theory model? Clearly not anything even close to things we have in the real world.
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>>12576933
Cool so 3+i is a real number.
>>
Just use a model of the reals where they're countable. Then we have no problems
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>>12577694
>Just use a model of the reals where they're countable.
Give an example of such a model.
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>>12577964
Just use the lowenheim skolem theory :)
>>
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>>12577632
>claiming something is rigorous
Rigor is never anything more than a matter of opinion. There is no objective standard of rigor. There's only a consensus view in constant flux.
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>>12578547
Try telling that to your analysis prof.
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>>12578576
Try telling Euler that his work isn't rigorous.
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>>12575256
i did it without infinities, give me my five bucks
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>>12578547
>Def 1.7: infty has properties of additive and multiplicative absorption
1) Why does zero, its reciprocal, only absorb multiplication and not addition?
2) Why is this schizo ass """axiom""" taken for granted without question? How retarded do you have to be to think "hurr 1+infty=0+infty"
>>
fucking hell
>>12578547
also, shouldn't [math]\frac{1}{n^{\pm\infty} } [/math] lead to a bifurcation for n>1?
Since
[math]\frac{1}{n^{+ \infty} } = 0[/math] and [math]\frac{1}{n^{- \infty} } = \pm \infty [/math]
>>
>>12579232
1) 1/0 is undefined and not equal to infinity so they aren't reciprocals.
2) It follows from the limit definition of infinity and isn't taken for granted.

>>12579428
I'm not sure what you mean and I think you have a sign error there.
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>>12579580
>I think you have a sign error
I knew it, damnit
>>
>>12579580
>It follows from the limit definition of infinity
your paper defines [math]\hat{\infty} [/math] as [math]\pm \infty [/math] without the additive absorbing property, but then defines both by using the same limit definition.
If additive absorption follows from this definition, how can [math]\hat{\infty}[/math] not have additive absorption?
>>
>>12579621
In this "quick" paper, it's due to Proposition 1.8. In my other non-quick paper where I disprove it without relying on an unproven proposition, I use [math]|\widehat\infty|=\infty[/math] as a workaround for the issue you describe.

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis
https://vixra.org/abs/1906.0237
>>
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Math is jewish subversion. You can't have sqrt(2) apples.
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>>12579621
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>>12573322
you have to define the real number you faggot, infinity is not a member of the mathematically defined set of "real numbers"
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>>12579981
>infinity is not a member of the mathematically defined set of "real numbers"
it should be, or else division is not closed over the reals
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>>12579988
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>>12578308
Requires infinite work
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>>12576155
If I create an axiom system ZFC+TPC, and then go on to base all my mathematics on it, then prove something known to be false in ZFC is that the equivalent of proving the TPC is false?
>>
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>>12580259
great question
>>
>>12579860
But you can have apple pi ;)



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