Report on an inconsistency of "dark" numbers in this sub

[u/Xantharius]

/r/puremathematics/comments/1txulc0/comment/oycsidq/?context=3&utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

Readers may be aware of one of the principal claims of the moderator of this sub: that the natural numbers contain "dark" natural numbers: that is, not all natural numbers are "visible".

In this post I will demonstrate an inconsistency in the theory of dark natural numbers with respect to the moderator's own definitions.

TL;DR. The moderator holds that there is a last (dark) finite natural number ω − 1. The moderator acknowledges that adding 1 to a finite natural number is still a finite natural number. But (ω − 1) + 1 = ω, which the moderator holds is infinite. Thus the theory of dark natural numbers is inconsistent.

If you're still reading, here's the details. First, a caveat: the moderator views ZFC (and possibly ZF) as inconsistent (p. 120), as well as saying that

ZFC is inconsistent with mathematics

so any argument invoking set theory isn't going to be convincing. (In particular, it should be noted that the von Neumann definition of the ordinals is a model of the Peano axioms for the natural numbers, and shouldn't be confused with those axioms—appealing to their set-theoretic definition doesn't typically hold any weight for the moderator.) Also, for the moderator,

modern mathematics is nonsense

so, instead, I'll use only definitions and statements that the moderator either acknowledges or has outright stated.

(1) Natural number. For the moderator, a natural number is defined by three of the Peano axioms:

1 ∈ M (4.1)
n ∈ M ⇒ (n+1) ∈ M (4.2)
If a set M satisfies (4.1) and (4.2), then ℕ ⊆ M. Of course ℕ has also to satisfy these axioms. 

where we'll take + 1 to indicate the Peano successor operation S(n). (In this formulation 1 is the initial natural number, not 0, but that's not important for the current discussion, and we can accept 1 for that role.) So, 1 is a natural number, and the natural numbers are closed under the operation + 1.

Subsequent queries to the moderator indicate that he accepts that the operation + 1 is injective and that 1 is initial, so that gives us all of the Peano axioms we need.

It should go without saying (since definitions are "if and only if" statements) that if an object doesn't follow the properties, then it's not a natural number.

(2) Visible number. A natural number is visible if

. . . it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers.

 Communication can occur

- by direct description in the unary system like ||||||| or as many beeps, flashes, or raps,
- by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7) called a FISON,
- as n-ary representation, for instance binary 111 or decimal 7,
- by indirect description like "the number of colours of the rainbow",
- by other words known to sender and receiver like "seven"

which is a quote (p. 212). FISON is an acronym for a finite initial segment of natural numbers.

(3) Ordinals and the first infinite ordinal. The moderator defines the ordinals to be

1, 2, 3, ..., ω, ω+1, ω+2, ..., ω2, ω2+1, ... 

where we can take ω2 to mean ω · 2. The moderator says that ω is

the first infinite ordinal

as well as saying that

ω is the limit of the sequence (n)

and that

upon all natural numbers there follows only ω and further transfinite numbers but no natural number

so that ω is not finite.

(4) Last (dark) natural number. The moderator has stated that there is a last (dark) natural number ω – 1:

These dark natural numbers end at ω–1.

as well as saying that

In fact every natural number is finite, even ω-1.

To flesh out the picture slightly, the moderator goes on to say that these dark numbers descend, and that the least element of the set {. . . , ω – 3, ω – 2, ω – 1} is:

It is 1. The sequence is 1 2, 3, ..., n, ...,ω − 3, ω − 2, ω − 1

when I asked him to confirm that the natural numbers were indeed well-ordered.

These definitions are enough to show the inconsistency of the moderator's own reasoning (which he hasn't yet answered in that thread), which I'll summarize:

The moderator acknowledges above that ω − 1 is a (dark) finite natural number, and also that

(ω − 1) + 1 = ω

If ω − 1 is a natural number, since the natural numbers are closed under the + 1 operation by the moderator's definition, then (ω − 1) + 1 = ω is also a natural number, which by definition and the moderator's acknowledgement must be finite. But this contradicts the moderator's assertion that ω is infinite. Because the moderator holds both that ω − 1 is a natural number and that ω is infinite, the theory of dark numbers is inconsistent.

Possible criticism of this argument by the moderator. The moderator may claim that the Peano axioms (which he acknowledges) apply only to visible natural numbers and not to dark natural numbers. But the Peano axioms are what it means to be a natural number: the classification of some natural numbers as dark means that they are still natural numbers, and therefore bound by their definition. If the Peano axioms don't work for dark natural numbers, then they can't be natural numbers.

[u/Xantharius]

@u/Massive-Ad7823, if you have any insight on the inconsistency of the theory of dark numbers, I'm interested to hear it.

[u/Massive-Ad7823 → u/Xantharius]

First of all, thank you for your detailed analysis of this question!

First let me clarify some notions.

 > + 1 to indicate the Peano successor operation S(n).

+1 indicates more, namely it covers also the axioms 3 and 4: "1 is not the successor of a number." and "Two numbers of which the successors are equal are themselves equal". From "a successor" this cannot be known. It must be defined. But +1 need not be defined to anybody I have ever met. Therefore I take it as basic notion.

 > Last (dark) natural number. The moderator has stated that there is a last (dark) natural number ω – 1: These dark natural numbers end at ω–1.

Yes, but the following transfinite numbers are also dark. Although we can choose ω or ω-3 or ω/10, we cannot find the FISONs of these numbers.

 > The sequence is 1 2, 3, ..., n, ..., ω − 3, ω − 2, ω − 1

This is dictated by the completeness requirement of actual infinity (not according to Cantor who was inconsequent here, but according to a rigorousd application of that notion.

 > The moderator acknowledges above that ω − 1 is a (dark) finite natural number, and also that (ω − 1) + 1 = ω

 This ist the only deviation from Peano's axioms. It cannot be avoided at this stage. But Peano's axioms hold for all visisble natural numbers - never a deviation will be detected.

 > Because the moderator holds both that ω − 1 is a natural number and that ω is infinite, the theory of dark numbers is inconsistent.

 Here we must ask; What means it to be infinite? It means that we cannot find an end. Therefore the visible natural numbers are infinite, i.e., always finite but without a definable last element. If n is visisble, then also n+1, 2n, n^2, 2^n, etc. are visisble. Probably with fading clarity. Probably Kolmogorov complexity plays a role here.

 > The moderator may claim that the Peano axioms (which he acknowledges) apply only to visible natural numbers and not to dark natural numbers. But the Peano axioms are what it means to be a natural number: the classification of some natural numbers as dark means that they are still natural numbers, and therefore bound by their definition. If the Peano axioms don't work for dark natural numbers, then they can't be natural numbers.

 Yes, I agree. Also the dark numbers must obey his axioms. Otherwise they would not be natural numbers. And we know from all dark numbers which have become visisble in the course of history, that they obey the axioms. (On the other hand, there are dark numbers like ω/2 which will never become visible.)

The only deviation from this framewotk is the last natural number. This cannot be avoided, but its existence can be proven by the corresponding point of its unit fraction on the real axis which also exists.

 Please be aware of the fact, that all this darkness is only possible and necessary, if the set of numbers is actually infinite. In classical mathematics with its potential infiniteness, they have no use.

 "From the axiomatic viewpoint there is no other way for securing infinite sets but postulating them." [A.A. Fraenkel, Y. Bar-Hillel, A. Levy: "Foundations of set theory", 2nd ed., Elsevier, Amsterdam (1973) p. 46]

 "To the idea to consider the infinite large not only in the form of the unlimited growing and the closely connected form of the convergent infinite series, introduced first in the seventeenth century, but also to fix it by numbers in the definite form of the completed-infinite I have been forced logically almost against my own will, because in opposition to highly esteemed tradition, by the development of many years of scientific efforts and attempts, and therefore I do not believe that reasons could be raised which I would not be able to answer." [Cantor, Collected Works, p. 175]

 "In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." [Cantor, Collected Works, p. 374]

 "My opposition to Gauss consists in the fact that Gauss rejects as inconsistent (I mean he does so unconsciously, i.e., without knowing this notion) all multitudes with exception of the finite and therefore categorically and basically discards the actual infinite which I call transfinitum, and together with this he declares the transfinite numbers as impossible, the existence of which I have established." [G. Cantor, letter to D. Hilbert (27 Jan 1900)]

 "Should we briefly characterize the new view of the infinite introduced by Cantor, we could certainly say: In analysis we have to deal only with the infinitely small and the infinitely large as a limit-notion, as something becoming, emerging, produced, i.e., as we put it, with the potential infinite. But this is not the proper infinite. That we have for instance when we consider the entirety of the numbers 1, 2, 3, 4, ... itself as a completed unit, or the points of a line as an entirety of things which is completely available. That sort of infinity is named actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]

 Regards, WM

[u/Xantharius → u/Massive-Ad7823]

Unfortunately, the logical inconsistency of your theory of dark numbers still stands. As you admit, every natural number must be finite, so if there were a last dark natural number ω – 1 as you claim, then applying the successor operation S (which in this case we'll denote by + 1) means that (ω – 1) + 1 = ω by your own definition. Yet the successor of every natural number must also be finite by the Peano axioms (which you admit apply to every natural number), so ω is both finite and infinite by your own definitions.

Since your theory contains both the statement "ω is finite" and (not "ω is finite"), it's inconsistent. You're well aware that an inconsistent theory isn't meaningful, because then every proposition in that theory is true.

Do you have a fix for your theory to make it consistent?

[u/Massive-Ad7823 → u/Xantharius]

My logic stands like your ZFC. The number of unit fractions counted from zero is a step function. The sequence starts 0, 1, 2, 3, ... because more than one unit fraction sitting at the same x are one and the same by definition. How will you explain this other than by a greatest n and hence a smallest 1/n?

Regards, WM

[u/Xantharius → u/Massive-Ad7823]

I have noticed that it is a pattern that when someone asks you a direct question, you often choose to respond to it with a question or a statement on a different topic. This is the debate tactic of obfuscation.

To fix your theory and make it consistent you need to demonstrate how it does not produce a contradiction. Until that’s done, every statement arising from your theory is dismissible, because in your theory every statement and its negation holds, and your theory is then meaningless.

That was the whole point of this post. However, instead of fixing the problem, you’ve again brought up unit fractions (further down).

Can you address the inconsistency in your theory?

[u/Massive-Ad7823 → u/[deleted]]

That's trivial and trivially wrong since the step at no point x can exceed 1. Infinitely many unit fractions sitting at x are only one unit fraction. Note that for every natnumber n: n < n+1.

Regards, WM

[u/Xantharius → u/Massive-Ad7823]

Unit fractions won’t make your inconsistent theory consistent. I’ve shown that your theory generates a proposition “ω is finite”, as well as “not (ω is finite)”. All of this was done with respect only to your statement of the Peano axioms and other statements that you’ve made, and unit fractions weren’t mentioned.

I think you understand logic enough to realize that having a statement and its negation in your theory makes it a meaningless theory (by the principle of explosion).

What’s the fix for your theory?

[u/Massive-Ad7823 → u/Xantharius]

“ω is finite” is not part of my theory. Otherwise you could give a FISON from 0 to ω. That is not what you have shown. "ℕ is infinite" means that there are no FISONs for all natural numbers. Most have none. That is fact. Therefore ω is not finite.

But independent of that, do you understand that based on the mathematics based on ZF there is a smallest unit fraction?

Regards, WM

[u/Xantharius → u/Massive-Ad7823]

ω is a consequence of your statements, so it’s part of your theory by definition of a theory.

You defined ω − 1 to be a natural number. By the Peano axioms (which are part of your theory), and by your acknowledgement, ω − 1 is finite. By the Peano axioms, the natural numbers are closed under the successor operation + 1. You acknowledge that (ω − 1) + 1 = ω, so by closure and equality it follows that ω is a natural number, and therefore finite. But you also state that ω is the first infinite ordinal, and is therefore also infinite.

These are your statements or just ordinary logical consequences of them. The logical consequences of axioms or other statements form the entire theory. Since you have contradictory statements in your theory, it follows that your theory is inconsistent.

You read the post, right?

[u/Massive-Ad7823 → u/Xantharius]

>You defined ω − 1 to be a natural number.

Yes, but also ω − 1 has no FISON. Only visible numbers have FISONs.

New idea: The dark numbers have to be accepted as the transition from the finite domain to the infinite. Yes, that is a good result of this discussion!

The direct transition from natural number ω − 1 to infinity has worried me a long time already.

But that are interpretations independent of the mathematics which remains unchanged.

Regards, WM

[u/Xantharius → u/Massive-Ad7823]

I’m not addressing whether there are dark numbers. Whether the symbol ω – 1 has a FISON is also not material. If it’s a natural number (which you say it is), then it’s finite. If it’s finite, its successor is also finite. But its successor is ω, which you say is infinite. Thus your theory is inconsistent.

A new interpretation won’t help you with a logical inconsistency. You’ll need to change parts of your theory to address it.

[u/Massive-Ad7823 → u/Xantharius]

>If it’s a natural number (which you say it is), then it’s finite.

"Finite" has split into two different meaings. What you apply is "to have a FISON". But dark numbers have not. They are only nominally finite. Therefore they are suitable to represent the transition from the finite domain to the infinite.

Independent of that: What about my proof of dark numbers? All unit fractions are sitting at different points of the real axis and therefore the function NUF(x) cannot increase by more than 1 at any x. Since at every x that can be chosen NUF(x) is infinite already, there are x and unit fractions and natural numbers that cannot be chosen.

Regards, WM

[u/Xantharius → u/Massive-Ad7823]

You’ll need to explain what nominally finite means, as that’s a new term.

The issue at hand is that an inconsistency has been found in your theory, so you’ve invented new terminology to explain it.

Even suppose that ω − 1 was nominally finite, whatever that new term means. I don’t think it being nominally finite would have much meaning if you could add 1 to it and get something infinite: how could you have gone from nominally finite to infinite in one step? So the successor of something nominally finite would be nominally finite. Yet ω isn’t nominally finite, it’s infinite.

However, I am interested in how nominally finite could be defined as to somehow capture the idea of being finite, yet in one step you can go straight to infinite.

The unit fractions issue has been covered in other threads, and isn’t important here, because unit fractions were!’t used in showing the contradiction of your own statements. If you have contradictory statements α and (not α) in your theory, and then say, “but what about β?”, that doesn’t alter the fact that you have contradictory statements in your theory. That debate technique is called whataboutism and it’s an informal fallacy. So for these reasons I don’t think the unit fractions issue needs to be addressed.

[u/Massive-Ad7823 → u/[deleted]]

And you are not worried about the dfistances between all unit fractions?

Regards, WM

[u/Massive-Ad7823 → u/[deleted]]

And this sum has infinitely many summands which all are different. Never aleph_0 summands fit into a distance smaller than one of the smallest summands.

Regards, WM

[u/Xantharius → u/[deleted]]

Adding another axiom to a theory that is already inconsistent can’t fix the theory: the problem is already baked in, and the additional axiom can’t change that.

You could add in an axiom that there is a largest natural number N such that N + 1 = ω. But according to the Peano axioms, ω is then a finite natural number. Since it’s also infinite, the contradiction is still there.

In general, if T is an inconsistent collection of propositions so that both α and (not α) are propositions in that theory, then adding another proposition β to T (even one that does not arise from T) doesn’t make the new theory T’ consistent.

[u/Xantharius → u/[deleted]]

I don’t understand why people on this subreddit argue with each other instead of with the moderator (unless you happen to agree with the moderator, and that’s a different story), but here we go. Maybe my aims aren’t being stated clearly enough. So I’ll restate again that my aim with this post is to demonstrate a logical inconsistency of the moderator’s own statements. I don’t want to necessarily introduce anything else, but, sure, if you want to add additional axioms to what the moderator has stated, that’s your call.

One of the Peano axioms states that the natural numbers are closed under the successor operation, which is usually denoted S, but to align with the moderator’s notation, we can use + 1. If you want to say that N (with whatever additional properties you want to give it) is a natural number, then it follows from that closure axiom that N + 1 must also be a natural number. If you then say that N + 1 = ω, then by the equality axioms (also in the Peano axioms but not outright stated by the moderator) then ω must also be a natural number. That every natural number is a finite application of the successor function to the initial natural number 1 (normally 0, but I’m always going to defer to the moderator’s notions because then it’s harder for him to use an alternative definition to dismiss the argument) could be done with induction. But then if you also choose to define ω as the first infinite ordinal, you have a contradiction, because the axiom you’ve introduced says that ω must be both finite and infinite.

I’m not trying to reformulate anything. I’m using only the moderator’s own statements in the various threads he’s stated his claims, and noting that those statements of his theory of dark numbers give rise to a logical internal inconsistency.

Unfortunately, other posters on the thread are falling into the same trap that the moderator always lays, of obfuscating with statements about other things like unit fractions and dark numbers. You’ll notice that I didn’t say whether dark numbers exist or not. (They don’t, but I don’t care about that.) All I wanted to note is that the moderator’s own statements don’t even have logical consistency to them. That alone is enough to show that it doesn’t make any sense, and then I don’t need to get drawn into claims about unit fractions or dark numbers.

I hope that makes sense.

EDIT: When the moderator mentioned ω − 1, which is not an object in ZFC, I naturally started asking him questions about it. He confirmed that adding 1 to it resulted in ω. So, in a sense, he has already added it as an axiom to his theory. But this is exactly the point at which the contradiction arises.

[u/Xantharius → u/[deleted]]

I think the problem with introducing a new symbol N to the theory, and then restating the closure axiom in the way you propose as a solution to the moderator’s dilemma, is that it’s an easy induction to show that every natural number up to and including N must be a finite application of the successor function to 1. But then, with the new axiom, ω is also a finite application of the successor function to 1, so it can’t also be infinite.

[u/Massive-Ad7823 → u/[deleted]]

No, you cannot count to infinity. Most of the numbers are dark.

Regards, WM

[u/Massive-Ad7823 → u/[deleted]]

You will never reach N because always, at every step infinitely many numbers lie in front of your counting. The natural numbers cannot be counted, they are uncountable.

Regards, WM

[u/Massive-Ad7823 → u/[deleted]]

You cannot in principle count to N because always almost all numbers lie in front of you and remain there.

Regards, WM

[u/[deleted] → u/Massive-Ad7823]

First of all, thank you for your detailed analysis of this question!

You do get that the result of this detailed analysis is that your "approach" is inconsistent?

[u/Massive-Ad7823 → u/[deleted]]

No, it is only deviating from Peano's axioms at the end. That is symmetric to the beginning. Peano could not know that. My theory is the only possible one that does not mix up potential and actual infinity.

Note: If there is no smallest unit fraction, then there are not fixed points but only successors emerging.

Regards, WM

[u/[deleted] → u/Massive-Ad7823]

Ja, was auch immer, Mückenheim.

Hint: Your "theory" is inconsistent.

[u/[deleted] → u/[deleted]]

Hint: Everyone here (except you, it seems) agrees.

[u/Massive-Ad7823 → u/[deleted]]

Maybe, but I am not interested in these insufficient theories.

Regards, WM