The continuum line can be said to have cardinal number aleph_1, — AgentTangarine

'the continuum' is probably most exactly defined as <R less_than>, but let's simplify here to just say it's R. It is the continuum hypothesis that its cardinality is aleph_1. It is not given or settled mathematics.

The two-dimensional continuum has cardinal number aleph_2, and the 3D continuum corresponds to alelph_7. — AgentTangarine

If by "the two-dimensional continuum" you mean RxR, then it is incorrect that its cardinality is different from the cardinality of R. If by "the 3D continuum" you mean RxRxR, then it is incorrect that its cardinality is different from the cardinality of R. For any natural number n>0, R^n has cardinality equal to the cardinality of R.

The subscripts are ordinal numbers and they correspond to the number of times an infinity of the infinite is needed needed to specify the elements — AgentTangarine

I don't know what you mean by "number of times an infinity of the infinite is needed to specify the elements" but the aleph notation is defined by transfinite recursion on the ordinals. For an ordinal k+1, aleph_k+1 is the least cardinal greater than aleph_k. For a limit ordinal L, aleph_L is the union of {aleph_k | k < L}.

He says he's out of his depth but persists by positing an "update". The update is nonsense updating nonsense. Meanwhile, he's not the least bit interested in actually studying the subject so that he could understand it and thus not serially broadcast his confusions and ignorance to other people.

The three books mentioned above are well beyond the preparedness of anyone who has not studied at least basic symbolic logic.

The best book for the layman about Godel's incompleteness proofs is:

Godel's Theorem - Torkel Franzen

The Godel sentence is a formula in the language of formal arithmetic. It an exact formula using only the symbols of symbolic logic and the symbols of arithmetic. It is an exact formal sentence about natural numbers. It is a very complicated formal sentence about natural numbers, but still it is a sentence about natural numbers. However, Godel discovered that certain formulas of arithmetic also are interpretable as sentences about formal theories. So the Godel sentence also happens to be interpretable as being true if and only if the Godel sentence is not provable in the formal theory in question. There is nothing suspect whatsoever about the rigor of the Godel sentence as a mathematical formula; not is it suspect whatsoever that the Godel sentence is true if and only if it is not provable in the theory in question.

Moreover, Godel's proof is purely about syntax and the proof itself does not rely on a notion of truth. That is the important way the Godel sentence is different from the liar statement. Whatever is problematic about the liar statement does not apply to the Godel sentence since the Godel sentence does not mention truth conditions nor is the Godel sentence self referential in the particulary problematic way of the liar statement. (It is only in an addition to the basic proof about syntax that we go on to note the the Godel sentence does happen to be true. And the conclusion from that is not "The Godel sentence is not provable in the system". Rather the conclusion is "If the system is consistent then the Godel sentence is not provable in the system".)

And Godel's proof can be formulated in primitive recursive arithmetic (purely finitistic combinatoric arithmetic) and with only intuitionistic logic. That would seem to be the minimum platform for mathematical reasoning. If there is any objection to Godel's proof then it would entail objection to even bare finitistic reasoning about natural numbers.

And fundamentally wrongheaded is the argument that Godel uses "false dichotomy" in the manner described in the first post. The proof is in intuitionistic logic, which does not provide "P or not-P" as a basis.

It is correct though that if one rejects the law of non-contradictions, such as certain systems of paraconsistent logic, then Godel's argument does not hold. Yes, one way out of Godel's theorem is to instead propose that mathematics be formulated paraconsistently. But that still does not impugn Godel's proof as much as it merely says there are alternative contexts in which the argument does not hold.

And beyond those points, the preponderance of posts above in this thread are an abysmally mangled misrepresentation of any aspect of Godel's proof. This is yet another example of abuse of the Internet to post serial misinformation and confusion. Posters who wish to present objections to Godel's argument should at least study the subject well enough to have informed and coherent notions about it.

What? .999... is not a sequence. It's a number. That number is 1.

The sequence is {<1 9/10> <2 99/100> <3 999/1000>...}

and its limit is 1.

Once more:

SUM[n = 1 to inf] 9/10^n = lim of {<1 9/10> <2 99/100> <3 999/1000>...} = 1.

EDIT: Corrected formulation of the sequence.

You keep replying past the simple straightforward points I made. Especially again as you go past my point that you haven't mathematically defined "reached". The proof that .999... = 1 does not require using an undefined notion of "reached", so it's not my problem that you keep wanting to bring it back into discussion.

Of course, if the sequence does not converge then it's another ballgame. But we easily prove that the sequence we've been talking about does converge.

You seem to be offering up diversions - infinitesimals, nonstandard analysis, sequences that don't converge - when the point is actually as simple as I've shown.

By the way, when I say that non-standard analysis is subsumed within classical mathematics, I mean non-standard analysis developed in ZFC. I'm not referring to IST axiomatically from scratch.

Specifically, we don't do that even in simple freshman calculus. I have never read an author say there is an "implicit reaching" (whatever that would actually mean, as you have not given a mathematical definition of 'series reaches').

And basic calculus does not use infinitesimals. It is in the context of ordinary classical mathematics that

SUM[n = 1 to inf] 9/10^n

is defined as

the limit of the sequence {<1 9/10> <2 99/100> <3 999/1000>...}.

And then with a finite proof we show that

the limit of the sequence {<1 9/10> <2 99/100> <3 999/1000>...} = 1.

No infinitesimals adduced and no undefined "reaching".

[Edit: I fixed the notation of the sequence.]

Of course non-standard has infinitesimals. (And non-standard analysis takes place itself in a larger environment of classical mathematics.) Though, I don't remember which, if any, infinitesimals are instantiated, since non-standard analysis, as I understand, non-constructivity is endemic to the proof of that infinitesimals exist. he proof of the existence . In any case, there is no notion of undefined notion of "reaching" needed for the classical method of a limit of a sequence such as discussed in this case.

What do you mean by 'instantiating infinity'. We instantiate the set of natural numbers in set theory, of course. And we instantiate the set of rational numbers in set theory, of course. The sequence is indexed by the set of natural numbers. And the range of the sequence is a subset of the set of rational numbers.

What exactly do you disagree with in what I just posted?

I've not seen it phrased that way, especially in a rigorous exposition in classical mathematics. Not even in freshman calculus. I don't know what writings in classical mathematics you have in mind.

There is an infinite sequence, with a finite description. And then there is a finite proof that the limit of that sequence is 1. No supertask.

after an infinite amount of operations — Shawn

There is an infinite sequence and there is the limit of that infinite sequence. There is no supertask - no performing an infinite number of operations - used.

what we understand intuitively can be understood in ways inconsistent with that in mathematics — Janus

Informally inconsistent, yes.

Anyway, when we move beyond child-level thinking that there must be involved a "reaching" and instead we study rigorous mathematics, then we understand that .999... = 1.

it seems we are not disagreeing about anything — Janus

To be in agreement, you'd have to agree that the limit of the sequence is rigorously defined.

there can be no actual infinite quantities of anything. — Janus

Of course, if one doesn't countenance infinite sets, then one might not countenance the classical notion of convergence to a limit. But that doesn't change that what ordinary mathematics means by '.999...' is not some kind of "reaching" but rather convergence to a limit.

if you added .9, .09, .009, .0009. .00009, and so on forever — Janus

And we don't do that.

By "instantiate an infinite series" I mean write it down as a full series, not in a shorthand form. I'm no mathematician, but that logical distinction is clear. — Janus

The word 'instantiate' has a certain meaning in mathematics. What you mean though - to type out in finite time and space individually all the members of an infinite set - is of course impossible. But that doesn't entail that the limit of an infinite sequence is not rigorously defined.

.

Of course, I don't begrudge adding a rubric 'absolute consistency' that way, though I like the term 'non-trivial' better.

A set S of formulas is non-trivial iff there is a formula P that is not a member of S.

With that definition, with a paraconsistent logic, a set of sentences can be closed under deduction while being inconsistent while being non-trivial. Pretty much another way of saying that EFQ does not obtain.

Godel does not use circular reasoning in the incompleteness proof. The proof can be given by means of assumptions no greater than finitistic combinatorial arithmetic and within intuitionistic logic. Contrary, to a post above that begins in abysmal ignorance with mentioning, in a contradiction in terms, a theorem that is not provable.

t is impossible to instantiate an infinite series — Janus

We instantiate them all the time. I instantiated the series in question earlier in this thread.

I'm here to please. — Olivier5

Then hurry up and take our lunch orders.

A set S of formulas is inconsistent iff there is a formula P such that both P and ~P are members of S.

As far as I know, that is the presumed mathematical definition.

paraconsistent systems avoid inconsistency by redefining it as other than (A & ~A), usually by adding a third truth value. — Banno

That is not my understanding, which is: Certain paraconsistent systems do not avoid inconsistency; rather they avoid explosion. But, yes, in the semantics, three truth values is one way. Also, in syntax one way is to use three values in truth tables and take derivation rules based on those truth tables. But, if I'm not mistaken, there can be dialetheistic semantics for paraconsistent systems.

Are you going to make a cogent argument as to how a series that is approaching one and could do so forever without actually reaching it is the "very same as I" — Janus

Again, it's not about a sequence "reaching" anything. '.999...' stands for the limit of a certain sequence. There is no "becomming" or "reaching". Simply, the limit is 1.

it approaches 1, but it never quite does become 1, though, does it? — Janus

Again, that reflects a fundamental misunderstanding of what '.999...' stands for.

Life is too short — Olivier5

Yeah, you really got that flippant dismissive thing down.

Anyway, better that life is too short than it be too long.

I tend to find SEP unreliable — Olivier5

I have not found problems (though, of course, no source is perfect). You asked about paraconsistency in context of engineering. I told you of a place where there is a real nice writeup (much more eloquent than, by your account, my own postings) about use in data systems; one could see how that could be adapted to an engineering context too.

So, my explanations are not eloquent enough for you, but you still are interested, but not interested enough to take a few moments from your too short life to read a professional account much better than I would write. Such a case you are.

I didn't say anything about truth values (semantics) for paraconsistent logics.

I am not expert, but, if I am not mistaken, the main point about a paraconsistent system is that it does not have EFQ. There is nothing stopping us from having premises that are a contradiction, and using a paraconsistent system to derive those premises trivially by the rule of placing a premise on a line. And those premises might be contentual axioms. So we would have theorems in contradiction with one another. And if the system has the rule of adjunction, then we can have the conjunction of the two contradicting premises. The important point though is that we can't use EFQ. But, of course, there are widely different kinds of paraconsistent systems, so I don't intend a complete generalization.

That's syntax (proof system). As to semantics, if I am not mistaken, not all paraconsistent systems accommodate dialetheism, but some do. Indeed the SEP article states that every dialetheistic approach must have a paraconsistent syntax . So, since the set of dialetheistic semantics is not empty, there must be paraconsistent systems (syntax) that accommodate dialetheism (semantics),

In a chart:

Exist. Paraconsistent syntax with dialetheistic semantics.
Exist. Paraconsistent syntax with non-dialetheistic semantics.
Exist. Dialetheism (which is semantics) with paraconsistent syntax.
Not Exist. Dialetheism (which is semantics) with non-paraconsistent syntax.

Paraconsistent logic does not allow contradictions; it does not allow (A & ~A) to be true. — Banno

If I'm not mistaken, that is incorrect as a generalization over all paraconsistent systems, as I mentioned above. As a rough generalization, paraconsistency does not "frown" on deriving a contradiction, and some paraconsistent approaches do not frown on having true contradictions. Rather, all paraconsistent systems don't have EFQ. (By saying that they don't have EFQ, I mean that they don't have "For all sentences P and Q, {P ~P} |- Q)".

Yes, I corrected myself. It's more complicated than I suggested. Indeed, you can see for yourself at the Stanford article. But it's still the case that paraconsistent logics do allow theorems of the form P & ~P.

What's intuitionistic about the 'inclusive or' in LEM? — Olivier5

I don't think it's particularly intuitionistic. Rather, it's that LEM (which is with inclusive-or) is eschewed in intuitionistic logic, while intuitionistic logic does accept LNC, so having LEM and LNC as different principles makes it convenient to describe intuitionistic logic relative to classical logic.

I mistakenly thought you meant 'bridge' in the sense of a connection between the two logic principles.

I have no idea about paraconsistent logic used in engineering. However, the Stanford article I suggested does talk about paraconsistent logic used in computing and especially for databases.

take out LNC to get paraconsistent logics — TonesInDeepFreeze

I need to correct that. Usually, paraconsistent logic is attained by not having EFQ. Taking out LNC would be something different. Nevertheless, I still think it is the case than having LNC and LEM as different principles bears upon paraconsistent logic though in a more involved way than I incorrectly stated.

I suppose this is the main advantage of dissociating the two. — Olivier5

Also, to be able to get intuitionisitic logic.

What do you mean by a "bridge"?

you wouldn't be too good at it — Olivier5

I certainly don't claim to be especially skilled in seeing into the minds of people who are ill-informed about the subject to know how they came to their misconceptions. Sometimes, though, I do sense at which point they got off track. And I did so soon enough with you too.

clue — Olivier5

It was not just a clue. It was a clear, concise, and precise statement.

I'm curious to see though whether you're going to continue with snide, petty, ill-premised misgivings against the person who gave you correct information.

We started this discussion two or three days ago though. — Olivier5

Oh, no, I didn't deliver your desired angle on the subject soon enough, even though what I did say was correct at every point while you persisted otherwise!

I gather that you have no idea what a nudnick you are being.

I adduced, entirely gratis, the point about LNC and LEM, because it was at that stage that I sensed it might bear upon your confusion. It's not my job to immediately anticipate what is confusing you about the subject and then to immediately warn you about those confusions. I gave you clear corrections, which is itself gratis, and then at other stages added more explanation also gratis. For that matter, there is even more about the subject I could add now, but, again, it's not my job to try to figure out why you are mixed up so that I choose just the right points customized for you.

Since P and ~P are mutually exclusive, what difference does it make whether the disjunction is inclusive or exclusive? — T Clark

P and ~P are mutually exclusive in classical logic, but not necessarily in other logics, especially paraconsistent logics.

To answer your question:

(1) It is important to not be confused as to which connective is actually used.

(2) With inclusive-or, we can take out LEM to get intuitionistic logic, and also, by not subsuming LEM within an LEM/LNC combo, we can take out LNC to get paraconsistent logics.

(3) The notions of LNC and LEM go back to antiquity, and have been critical in the discussion of logic through the centuries, so to suddenly say that LEM now means something different would be quite confusing.

[Edit: I should not have written, "We can take out LNC to get paraconsistent logics."]

You could have said a long time ago: "you must mean the LNC, because the LEM does not actually rule out contradiction." — Olivier5

I don't know what you mean by 'a long time ago' relative to the duration of our exchanges. But many posts ago, I wrote:

the 'but not both' clause for exclusive or is demanded by the law of non-contradiction: ~(P & ~P). — TonesInDeepFreeze

You're blaming me for the fact that you didn't bother to read what I posted.

consistency is unprovable within the system — tim wood

And unprovable from other systems of a certain kind.

to say that a system is unprovable is not to say that processes within the system are not provable, or in any way inconsistent. 2+2=4 being a simple example of a true and consistent expression from within mathematics. — tim wood

Systems are not things we look at for being probable or unprovable. Maybe you meant this: That a particular system can't be proven consistent by certain means does not entail that the system is inconsistent.

you're not a very eloquent writer — Olivier5

My posts aren't models of eloquence, but they are, for the most part, articulate and more precise about technical matters than normally found in a casual context such as this forum.

After a series of clearly correct explanations by me, and over a course of your stubbornness, finally those explanations provided you with understanding why you were incorrect to begin with, and also provided you with context of other notions that are important to the topic. But the very fact that I provided you with ample explanations leads you to gripe that they were not eloquent enough for you and too ample.

It would take you ages. — Olivier5

Yes, an explanation why your notion of an advantage of a certain elegance is wrong headed would be longer than a couple of lines, because your notion is wrong headed in so many different ways.