Anonymous at Thu, 7 Mar 2024 02:38:52 UTC No. 16060774

>>16060767
There are a few arguments for finitism that I see (all of which are really dumb).

The two most common I've seen people argue are:
1) Numbers are all representations of physical quantities (matter, energy, time, quantity of an object, etc.) and as a result only a finite number of them can exist. This is dumb and people who believe it are dumb.

2) They believe that all of the reals must actually be computable (and thus are countable in size) and can not conceptually handle the concept of an uncountable set.

Anonymous at Thu, 7 Mar 2024 02:40:54 UTC No. 16060775

>Dedekind cuts are invalid because... they just are, okay? infinity is not a place
https://youtu.be/LSWIFXP2r14

Anonymous at Thu, 7 Mar 2024 02:47:14 UTC No. 16060782

>>16060775
Top kek

Anonymous at Thu, 7 Mar 2024 03:49:13 UTC No. 16060824

>>16060774
Neither of which really home in Wildberger's take. The second is closer, but only by necessity of wanting "equivalence class of sequences" to mean something stronger than "potentially exists if you define a function". Potentiality and convergence are useful, but he disagrees with the assumptions made by baking them into the number system.

Side note: even Wildberger says [math]0.\overline 9 =1[/math], schizos who hate geometric series need not apply.

Anonymous at Thu, 7 Mar 2024 03:57:21 UTC No. 16060831

>>16060824
I don't know Wildeberger's presentation of finitism, but it wouldn't surprise me to hear that it's more reasonable than most I see here and on social media.

Can you explain what his problem is with the notion of equivalence classes for proving the reals are uncountable? It's definitely counter intuitive to think about the rationals being dense, and yet there are still an uncountable number of non-rational reals between any two particular rationals you choose.

The most convincing arguments I've seen really don't construct the reals as an uncountable set, but demonstrate that it can't be possible for the reals to be countable. Cantor's argument about equivalence classes and the set of all numbers constructed with one sample from each equivalence class in [0,1] is fancy, but it seems more useful to prove things with respect to the measure of real intervals rather than their countability/uncountability.

Anonymous at Thu, 7 Mar 2024 04:29:23 UTC No. 16060853

>There is no integer [math]n \in \mathbb{Z}[/math] with [math]n^2 = -1[/math]? Just make it up, bro! Let's call it a dark integer.
Is this a valid way of doing math?

Anonymous at Thu, 7 Mar 2024 05:38:55 UTC No. 16060928

math is just human imagination run rampant and can be whatever you want it to be :)

Anonymous at Thu, 7 Mar 2024 05:43:18 UTC No. 16060932

>>16060853
i isn't an integer. It is a basis vector in the R2 representation of the complex plane.

It's actually an incredibly practical concept that basically corresponds to a 90° angle of rotation.

Anonymous at Thu, 7 Mar 2024 07:08:47 UTC No. 16061026

>>16060767
>They can be defined constructively using type theory
All definitions suck and they're not equivalent with each other. It's reasonable to reduce the usage of real numbers to a minimum for that reason alone, and possibly to nothing.

Anonymous at Thu, 7 Mar 2024 07:19:33 UTC No. 16061040

>>16060775
The argument is that real numbers codify the concept of "computation with arbitrary precision" alongside numbers

Anonymous at Thu, 7 Mar 2024 07:25:32 UTC No. 16061046

>>16060831
NTA but Wildberger's presentation of finitism is surprisingly childish. He dresses it up like it's insightful and tries to handwave but his real argument is that real numbers aren't real because you can't write them down or define a general algorithm that can produce the digits of any given real number. Wildberger has never given a solid argument against real numbers. His central complaint is "real numbers are too weird and that makes me uncomfortable." It's nice that someone's questioning foundations, but he really is just entirely off-base.

Anonymous at Thu, 7 Mar 2024 07:42:01 UTC No. 16061072

>>16061046
>can't write something down or define a general algorithm
If this is his main complaint then I have bad news... All math and computer science is not real.

Anonymous at Thu, 7 Mar 2024 08:18:45 UTC No. 16061112

>>16060774
>its dumb because i say it is
ok retard

Anonymous at Thu, 7 Mar 2024 14:10:15 UTC No. 16061353

>>16061046
You'd think he'd accept the algebraics, but no - he goes on to do trig on the Cartesian plane of rationals, and avoids Euclidean distance by always using the square of distance.
So (0,0)->(0,1) has a distance of 1, but (0,0)->(1,1) has a "squaracity" of 2.

Anonymous at Thu, 7 Mar 2024 14:30:02 UTC No. 16061371

>>16061112
No, it's dumb because it is far less capable than conventional math while being no more consistent and having no real arguments in its favor.

There's no reason to believe it's true, and it's not even any more useful than the conventional construction of set theoretic mathematics.

Cult of Passion at Thu, 7 Mar 2024 15:25:39 UTC No. 16061427

>>16060774
>physical quantities
Units of measure*.

Anonymous at Thu, 7 Mar 2024 15:44:10 UTC No. 16061450

>>16061026
Cauchy reals (as a setoid, with an explicit modulus of convergence) are the ones useful for applied math and therefore the correct choice.

Anonymous at Thu, 7 Mar 2024 19:11:54 UTC No. 16061669

>>16061450
And then you can do reverse mathematics by just restricting induction and comprehension.

Anonymous at Thu, 7 Mar 2024 21:03:47 UTC No. 16061799

>>16061072
What if it's kinda like how miasma was considered the blame for disease, when it was in fact the pathogens that caused the infection? And all the while terrain theory tries to poke holes in those arguments and assertions. Perhaps marriages of concepts require some borderline paradoxical thinking nestled between adaptive constants for ease of understanding? Now what might those be called again? And what would that mean for computers? Well we only understand and measure what we see, and if we build and work within those confines without testing and questioning, then we can't progress forward. Never take any field as fully established. In 100 years of advancement you might feel foolish.

Anonymous at Fri, 8 Mar 2024 04:37:06 UTC No. 16062307

>>16061799
>only understand what we see
Where did you "see" this thought? Or where did you measure that "we only understand and measure what we see"? Empiricism is self-refuting, anon. You can't base your view of reality only on empirical observation because empirical observation itself isn't sufficient to show this.

Anonymous at Fri, 8 Mar 2024 04:47:09 UTC No. 16062319

>>16060774
What do they have to say to stuff like the x and y axes on a graph being infinitely extensible? Are we supposed to artificially cap them or something?

Anonymous at Fri, 8 Mar 2024 05:03:40 UTC No. 16062336

>>16062319
If I had to try and make an argument from that perspective (I'm not a finitist) they would probably say something along the lines of "there's no particular x or y after which you have to stop your axes, but looking at f(x) over the infinite set of x values doesn't make sense."

They'll never really explain why it doesn't make sense, or how the same process that works with arbitrarily large finite sets suddenly stops working when the set is infinite. If they believed in modern set theory they could maybe point out that the countable intersection of closed sets is not necessarily closed (as an example), but they generally don't engage at all with the question.

Anonymous at Fri, 8 Mar 2024 09:00:28 UTC No. 16062553

>>16062319
>What do they have to say to stuff like the x and y axes on a graph being infinitely extensible? Are we supposed to artificially cap them or something?
No need to cap. You already pay the price by graphing them. You'll never be able to graph the whole thing anyway.

Anonymous at Fri, 8 Mar 2024 09:13:27 UTC No. 16062557

>>16062336
>They'll never really explain why it doesn't make sense, or how the same process that works with arbitrarily large finite sets suddenly stops working when the set is infinite.
Two things:
1. Any "arbitrarily large" set is infinitely small compared to an actual infinite set, as there are infinitely many sets bigger than it, but only finitely many that are smaller. The very admission of a similarity between an arbitrarily large set and an infinite set shows how shallow people's conception of infinity really is, taking it to mean basically "just big enough for me to not care". And they're right, finitists are just more honest about it, see the next point.
2. You can go as large as you're willing to sit out and calculate. You'll reach any arbitrarily large finite set given an arbitrarily large amount of time, but you'll never reach an infinite set. Very large sets are already quite hard to actualize, and the larger the set, the harder it becomes, and past a certain point it becomes virtually impossible, hence the previous point. But you have no way to work with the infinite other than by what can be generated through finite description, even in principle.

Anonymous at Fri, 8 Mar 2024 09:47:04 UTC No. 16062578

>>16061046
>questioning foundations
>entirely off-base
i get the feeling that there is a joke there

Anonymous at Fri, 8 Mar 2024 09:50:18 UTC No. 16062582

>>16061353
not the guy you responded, but is that really so?, boy that's funny

Anonymous at Fri, 8 Mar 2024 09:57:20 UTC No. 16062584

>>16062553
>pay the price
is the a graph tax now?, i have indeed notices that a specially weird argument certain finitist's put forth is that they don't want tax money to be spent in infinitary math, boy are they weird

Anonymous at Fri, 8 Mar 2024 13:58:25 UTC No. 16062764

>>16062557
Ah, so it's a computation forward mindset. I understand.

I disagree with both of your points, and I want to try to explain why.

1). When you look at an arbitrarily large set, and you show that the difference between the mapping on the finite but large set, and the infinite set (countable, uncountable, whatever) gets arbitrarily small then you don't need to "calculate it" beyond the point of you caring. If the rule stays consistent and works for any epsilon > 0, it will work in the limiting case as epsilon approaches zero and the set grows to infinite size. The way I like to think about this is via a Fourier series. Generally, the first hand full of terms contribute the "bulk" of the information in whatever convergent set mapping you are looking at. If the set mapping converges as it grows to infinite size, it has to be the case that the elements grow smaller and smaller and smaller in their contribution. That's the whole reason why convergent results on infinite sets or infinite series can function and why many infinite set functions/infinite series don't converge.

2) This also assumes you are calculating something and shows a very "computational" approach to mathematics. If you can demonstrate that your contributions from including these elements grow smaller and smaller and that the limit at infinite size exists, then you're done. You don't need to fret about literally doing the calculations like you're some freshman computer science major. Use your brain a little bit man.

Anonymous at Fri, 8 Mar 2024 15:28:40 UTC No. 16062897

>>16062764
>1).
There's more to infinite sets than converging sequences. Hell, there's more to real numbers than converging sequences.
If you applied the same idea to real functions, that finite portions of the input should only affect a finite of the output, you'd end up with some sort of Brouwerian continuity. Discontinuous functions already break that property.
This doesn't answer anything that I've said at all.

>2) This also assumes you are calculating something and shows a very "computational" approach to mathematics.
To engage with mathematics is to compute. You are given a set of axioms per field, and you use finitely many applications of them to reach conclusions. I'm not a freshman computer science major, and you're not an highschooler either. "Calculation" is not limited than highschool algebra. The only way to engage with infinities is through finite procedures.

>If you can demonstrate that your contributions from including these elements grow smaller and smaller and that the limit at infinite size exists, then you're done. You don't need to fret about literally doing the calculations like you're some freshman computer science major. Use your brain a little bit man.
Yes, I don't (necessarily) doubt that you have a procedure to generate rational numbers that grow towards a limit, I'm just more honest than you about its significance.
The moment you pass from working with exact rational numbers to approximate real numbers, you devalue your work a little. You give yourself stronger premises to derive your results more easily. Fair, but some people are more interested in stronger results under weaker premises.
Then you also tell yourself false narratives, as if pushing symbols on a piece of paper grants you control over actual infinities. That's delusional and objectionable, and not at all what you're actually doing.
Finitists object to either the latter or both things.

Anonymous at Fri, 8 Mar 2024 15:45:28 UTC No. 16062915

>>16062897
>Then you also tell yourself false narratives
>implying the human mind created in the image of God lacks the ability to operate with created infinities
This is more delusional than even formalism.

Anonymous at Fri, 8 Mar 2024 15:49:13 UTC No. 16062917

ZFC/finitism is a false dialectic. Type theory is where it's at. Much cleaner definitions and it is actually constructive. Finitists ignore it like it doesn't exist.

Anonymous at Fri, 8 Mar 2024 15:57:00 UTC No. 16062931

>>16061046
>define a general algorithm that can produce the digits of any given real number
that's a strong objection though

Anonymous at Fri, 8 Mar 2024 16:01:09 UTC No. 16062937

>>16062931
Undecidability/not being computable isn't an indication of not being "real". There is no general algorithm for producing the 'correct' laws of logic. It's a very weak and inconsistent objection. If the same standards are applied to other things in math, math itself would not be real under such reasoning.

Anonymous at Fri, 8 Mar 2024 16:11:24 UTC No. 16062960

>>16062897
> There's more to infinite sets than converging sequences.

This is clearly true, but you're missing the point. When finitists claim that operations on infinite sets are fundamentally illegitimate, you have a problem when there are whole classes of operations on infinite sets that are well behaved. The claim of finitism isn't "some infinite mathematical objects are legitimate and some are not." That claim is one that is widely supported in conventional mathematics and why people study things like absolute convergence, convergence under metric norms, monotone class convergence etc.

The point I'm making is that there clearly are well behaved infinite set procedures that produce legitimate mathematical mappings without the need to directly compute all of the elements. That's enough already to know that strict finitism as a formalization of math has problems. Then we're just fighting over degree and which kinds of infinite set operations are illegitimate and it's really more of a question of real analysis and set theoretic convergence than number theory.

> To engage with mathematics is to compute.

I fundamentally disagree with this framing of mathematics. There are sub-fields within mathematics that are devoted to computation (e.g., algebra, number theory) and there are sub-fields of mathematics that are devoted to other concepts entirely.

As an example, measure theory is a foundational field of mathematics from which a ton of different disciplines and mathematical pursuits emerge. Measure theory has elements which are computational (e.g., actually computing a particular measure of a set, or a functional integral in a measure space on a set), but has major sections of it devoted to determining whether or not a particular set, collection, or function is even measurable. This is not a computational pursuit where we are manipulating point values or measures. It is looking at whether the measure space itself is consistent.

Anonymous at Fri, 8 Mar 2024 16:54:48 UTC No. 16063040

>>16062937
This, the busy beaver function is uncomputable but outputs natural numbers, which even Wildberger admits are real. Actually wait, in the past he's even argued that natural numbers with enough digits aren't "real" because they can't fit in the observable. I think he's just kind of retarded and inventing justifications post hoc.

Anonymous at Fri, 8 Mar 2024 16:55:54 UTC No. 16063044

>>16063040
*observable universe

Anonymous at Fri, 8 Mar 2024 16:57:15 UTC No. 16063046

>>16063040
Do you know if he believes in God? What's his stance on formalism vs math being in the divine mind?

Anonymous at Fri, 8 Mar 2024 17:01:24 UTC No. 16063060

>>16060767
The main problem is that they are not mathematically rigorous. Real numbers have never been given a full definition and no general method was given to determine if something is or isn't a real number.
> Is there a real logical argument for finitism?
Yes. Simply ask the infinitist for a proof or definitions of the transfinite concepts they use. Watch them scramble and deflect.
>>16060775
They are invalid because a general subset of the rational numbers, as a concept, has not been defined.
>>16060831
> Can you explain what his problem is with the notion of equivalence classes for proving the reals are uncountable?
His problem has always been the fact that the notion of a real number is not well-defined. It's based on a fantasy and wishful thinking.
>>16061046
His presentation is lacking sometimes but the core of his arguments is clear.
>>16061072
Not true.

Anonymous at Fri, 8 Mar 2024 17:03:53 UTC No. 16063072

>>16062319
Finitists are ok with graphs being infinitely extensible.
Wildberger is an ultrafinitist, so he wouldn't be ok with that.
>>16062336
> If I had to try and make an argument from that perspective (I'm not a finitist) they would probably say something along the lines of "there's no particular x or y after which you have to stop your axes, but looking at f(x) over the infinite set of x values doesn't make sense."
Not true. Q is an infinite set and looking at f(X) over Q makes sense to a finitist (me).

Anonymous at Fri, 8 Mar 2024 17:05:49 UTC No. 16063074

>>16062917
Define a real number in type theory then, and prove that arithmetic on them is well-defined and has the desired properties.
Pretty sure you have no idea what you're talking about.

Anonymous at Fri, 8 Mar 2024 17:06:23 UTC No. 16063076

>>16063060
>Real numbers have never been given a full definition
https://unimath.github.io/agda-unimath/real-numbers.dedekind-real-numbers.html
Full formal definition or reals in a _constructive_ system.

Anonymous at Fri, 8 Mar 2024 17:06:50 UTC No. 16063080

>>16062937
>Undecidability/not being computable isn't an indication of not being "real".
Not having a definition, and only being based on feelies and vibes, is a pretty good indication of not being real. Such is the state of real numbers.

Anonymous at Fri, 8 Mar 2024 17:09:12 UTC No. 16063084

>>16062960
>When finitists claim that operations on infinite sets are fundamentally illegitimate, you have a problem when there are whole classes of operations on infinite sets that are well behaved
Like the power set? Remind me what's P(w)? Is it w_1, w_2, w_13123123 or something else entirely? It's not that they're ill-behaved, but rather they're undefined, like infinite sets themselves. Mathematicians just assume they're well-defined because they want them to be.

Anonymous at Fri, 8 Mar 2024 17:10:26 UTC No. 16063088

>>16063076
How is a "mere predicate" defined?
Pretty sure this is just more of transfinitist nonsense.

Anonymous at Fri, 8 Mar 2024 17:11:12 UTC No. 16063090

>>16063060
>no general method was given to determine if something is or isn't a real number.
Only a problem in set theory where whether [math]x \in \mathbb{B}[/math] holds is a valid question for all sets. In type theory it is just applicable only to what is _provably_ a real number, then there is a judgement [math]x : \mathbb{R}[/math]. If you want to show some object "is" a real, you have to provide a point in [math]\mathbb{R}[\math].

Anonymous at Fri, 8 Mar 2024 17:13:18 UTC No. 16063096

>>16063088

Anonymous at Fri, 8 Mar 2024 17:14:19 UTC No. 16063099

>>16063046
Not sure if he actually believes in god but he sure does talk about about him. He loves to say things like "you can't do infinitely many things only god can do that, saying you can write down a real number is arrogant. humans arent god."

Anonymous at Fri, 8 Mar 2024 17:15:21 UTC No. 16063101

>>16063090
So according to you, in type theory a real number is that which is provable to be a real number. Now what is *that object* x that we prove/make a judgement x: R about? What possible forms does it take? Next, how do we determine whether it's provable or not if x: R?

Anonymous at Fri, 8 Mar 2024 17:15:22 UTC No. 16063102

>>16063099
If humans are made in the divine image, and can actually access reality directly via our intellect, then there is no issue with "big" numbers that hold more information than the universe. I truly do not see his point.

Anonymous at Fri, 8 Mar 2024 17:16:33 UTC No. 16063103

>>16063102
Human mind being finite and limited compared to god's mind is a common trope in many religions.

Anonymous at Fri, 8 Mar 2024 17:17:34 UTC No. 16063106

>>16063090
Suppose I have found two real numbers x: R and y:R. How do I determine whether x=y?

Anonymous at Fri, 8 Mar 2024 17:21:29 UTC No. 16063114

>>16062960
I agree with you that by arguing against a strawman and by ignoring the substance of counterarguments, finitism ends up looking pretty dumb.

>The claim of finitism isn't "some infinite mathematical objects are legitimate and some are not."
Indeed, their claim is actually threefold: that actual infinite objects don't exist, that when mathematicians tell themselves they are "working with infinite objects" they are actually doing something else, and that that something else is finitary and less useful by being something more intentionally finitary. An example of that diminished utility is the plethora of pathological objects, constructable or not, that arise in infinitary mathematics: even your own formal systems are telling you that something's not right.

>I fundamentally disagree with this framing of mathematics. There are sub-fields within mathematics that are devoted to computation (e.g., algebra, number theory) and there are sub-fields of mathematics that are devoted to other concepts entirely.
Not what I said at all. What I said is:
>You are given a set of axioms per field, and you use finitely many applications of them to reach conclusions.
This applies to real analysis all the same.

Anonymous at Fri, 8 Mar 2024 17:22:03 UTC No. 16063116

>>16063101
All types are defined inductively/synthetically. [math]x : \mathbb{R}[/math] is not a proposition in the system itself like it is in set theory. It's a meta-level judgement that something _is_ already a real number. It's necessarily a point in [math]\mathbb{R}[/math] witnessing that [math]\mathbb{R}[/math] is inhabited.
>>16063103
Maybe in Islam it might be a problem with their ultimate separateness of humanity from God, but not in Christianity where we are in God's image.
>>16063106
You have to show they define the same Dedekind cut according to the rules you constructed [math]\mathbb{R}[/math] with. In general it is undecidable.

Anonymous at Fri, 8 Mar 2024 17:24:12 UTC No. 16063120

>>16063084
> It's not that they're ill behaved, but rather they're undefined, like infinite sets themselves.

Do you not see how the second part of your sentence is problematic when there are infinite sets that are defined? There being infinite sets that are not well defined does not mean that all infinite sets are undefined. You are jumping to a strange conclusion based on a negation of something that nobody beleives. Nobody believes "the set of all infinite sets" (as an example) is something that is well defined. There are infinite sets that are undefined, there are infinite sets that are defined.

Anonymous at Fri, 8 Mar 2024 17:24:46 UTC No. 16063123

>>16063116
So equality is undefined in your system of "real numbers". Very cool.

Anonymous at Fri, 8 Mar 2024 17:26:30 UTC No. 16063128

>>16063120
>Do you not see how the second part of your sentence is problematic when there are infinite sets that are defined?
No. Some particular infinite sets have particular definitions and that's fine. There's no general definition of what is meant generally by an "infinite set". That's what I meant.
The set of rational numbers is perfectly fine, even to a finitist. A general subset of rational numbers is not, because that's not well-defined.

Anonymous at Fri, 8 Mar 2024 17:26:42 UTC No. 16063129

>>16063120
Even natural numbers have nonstandard models.

Anonymous at Fri, 8 Mar 2024 17:27:31 UTC No. 16063131

>>16063129
This is false.

Anonymous at Fri, 8 Mar 2024 17:28:46 UTC No. 16063134

>>16063131
https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic

Waiting for you to mention second-order arithmetic next.

Anonymous at Fri, 8 Mar 2024 17:30:08 UTC No. 16063138

>>16063134
The "proof" that such nonstandard models exist relies on illegitimate and undefined transfinite techniques. As such, the proof is invalid.

Anonymous at Fri, 8 Mar 2024 17:30:17 UTC No. 16063139

>>16063123
Equality is defined in type theory for all types, reals have absolutely nothing to do with it. Being "defined" is not the same thing as having a general algorithm for equality within a specific type. There are statements even in finite group theory that don't have a general algorithm that decides the truth of them.
>GROUPS ARE NOT REAL CAUSE there's NO ALGORITHM TO DECIDE WHETHER TWO OF THEM ARE ISOMORPHIC
Also the halting problem is completely constructive, yet there is no algorithm for deciding when two finite programs are "equal" or when a finite program halts on a certain input.

Anonymous at Fri, 8 Mar 2024 17:30:26 UTC No. 16063140

>>16063102
A lot of his arguments seem to imply that anything you cannot understand completely you should not bother trying to understand partially. Yes, I cannot hold a real number perfectly with all its digits in my mind, but I know many things about it, can manipulate it according to rules I am capable of understanding, and in a pinch, write it down explicitly to whatever precision is needed for any practical purpose.

Anonymous at Fri, 8 Mar 2024 17:30:26 UTC No. 16063141

>>16063103
>many religions.
such as?, name three

Anonymous at Fri, 8 Mar 2024 17:31:03 UTC No. 16063143

>>16063141
NTA, but atheism, islam and judaism come to mind.

Anonymous at Fri, 8 Mar 2024 17:32:36 UTC No. 16063145

>>16063143
He said name three. I'll admit that atheism and islam are different, but atheism=judaism and judaism=islam.

Anonymous at Fri, 8 Mar 2024 17:33:12 UTC No. 16063146

>>16063145
Calvinism/Catholicims with its idea of created grace and inability to participate in uncreated divinity.

Anonymous at Fri, 8 Mar 2024 17:33:46 UTC No. 16063147

>>16063138
>proof by "nuh-uh"
must be easy to be a such a limited being

Anonymous at Fri, 8 Mar 2024 17:40:52 UTC No. 16063157

>>16063146
Ok now name 5-7 sects of Christianity which reject the trinity

Anonymous at Fri, 8 Mar 2024 17:51:46 UTC No. 16063172

>>16063129
Well yes, you could have a non-standard model for really anything mathematical.

That's fine. I don't have a problem with people believing in non-standard model or even with people believing in finitism. I just personally don't see the arguments for it being particularly interesting or convincing.

Anonymous at Fri, 8 Mar 2024 17:55:59 UTC No. 16063178

>>16063128
> There's no general definition of what is generally meant by an infinite set.

In some sense this is true, and why people who work in measure and topology go to great lengths to define things like sigma-algebras or metric topologies to keep things more meaningful. Even in this case there are infinite sets that we know must exist that are intractably complicated to describe via some finite number of rules. That is genuinely a challenging concept.

In another sense, I don't think I agree. All one means by an "infinite set" is that it is not possible to satisfy the definition of the set with only a finite number of elements. It doesn't tell you anything more about its structure. It just tells you that it is not possible to answer the implicit problem to which the set is the solution with only a finite number of answers.

Anonymous at Fri, 8 Mar 2024 18:15:44 UTC No. 16063211

>>16060853
yes and no
yes in the sense that, you can make up any numbers with whatever properties you want,

no in the sense that you must prove their logical consistence, ie: you can always trace their existence back to axioms(this proof does exist for complex numbers) and they don't create contradictions

hyper-real numbers are another interesting extension of the number line.
as are quaternions, etc...

Anonymous at Fri, 8 Mar 2024 19:00:50 UTC No. 16063288

>>16062917
I took a 2 year break from HoTT to read Hatcher
Still not done with it since I've been busy with other things

Anonymous at Fri, 8 Mar 2024 19:03:26 UTC No. 16063299

>>16063288
I've only recently gotten back into it too. Took a break from all math for a while and it's interesting to start again.

Anonymous at Fri, 8 Mar 2024 19:10:00 UTC No. 16063308

>>16063157
not him, but islam's under some technicalities one such sect
https://www.youtube.com/watch?v=U5c-d1pNnvs
https://www.youtube.com/watch?v=j-lAekcXRpw

Anonymous at Fri, 8 Mar 2024 20:17:42 UTC No. 16063417

>>16063106
what do you mean by "found"? like you found them on the beach and picked them up?

Anonymous at Sat, 9 Mar 2024 06:05:57 UTC No. 16064494

>>16063417
He's being cheeky.
>suppose I found
It's a false assumption in his worldview.

Anonymous at Sat, 9 Mar 2024 06:15:24 UTC No. 16064502

>77 posts
>20 IPs

Anonymous at Sat, 9 Mar 2024 16:15:45 UTC No. 16065119

bump

Anonymous at Sat, 9 Mar 2024 16:16:33 UTC No. 16065120

>>16064502
One glorious samefag

Anonymous at Sun, 10 Mar 2024 11:06:19 UTC No. 16066250

Bump.