You need to first learn basic symbolic logic. I recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague and Mar. Then basic set theory in which the systems of logic (including paraconsistent logic) are formalized. I recommend 'Elements Of Set Theory' by Enderton. Then mathematical logic, which is a deeper study of logic. I recommend 'A Mathematical Introduction To Logic' by Enderton, supplemented with 'Introduction To Logic' by Suppes for his discussion of mathematical definition, which is the very best discussion of that subject I've seen.

That will give you a clear and rigorous understanding as opposed to flitting about among various bits from improperly edited Wikipedia pages, taken out of context or even grossly misunderstood and then irresponsibly misrepresented by you on this Internet forum.

That post and most of what is said in the poster's following posts is terribly ill-premised.

Classical logic itself does not result in contradictions. Rather, adding certain non-logical axioms results in contradictions. For example, classical first order logic is consistent, but if we add an axiom schema of unrestricted comprehension (which is a set of non-logical axioms) then we get the contradiction known as 'Russell's paradox'.

A logical principle is one that is true in all models. A non-logical principle is one that is not true in at least one model.

It is fatal mistake not to understand that difference between logical axioms and rules (the bare bones of deduction) and non-logical axioms or rules.

Classical logic, which includes the explosion principle, is consistent. It is nonsense to claim that classical logic can be consistent only if it eschews the explosion principle.

There are many other terrible misconceptions stated by the poster, but at least it's a start to point out the lack of distinction between the logic itself and what can be derived using the logic from additional non-logical axioms.

I don't know what you mean by 'Newberry's [the poster's] parameter' or 'only staying within' it.

As I understand, it is suggested that we could add have a theory PA$ (still in the language of PA) that is PA with an added axiom schema:

For all formulas F,

provable(#(F)) -> F

/

I deleted my questioning whether that is a properly defined set of axioms. Yes, it looks properly defined and recursive:

We take any formula F, then calculate its Godel number, say n, then we have as an axiom:

provable(n) -> F.

But now I wonder whether U is a well defined, let alone recursively, axiomatized theory if it includes this axiom schema:

For all formulas F,

provable_U(#(F)) -> F

There you've defined the axioms of U in terms of the theorems of U, which, if I'm not mistaken, is circular, not well-defined.

We say, "provable_U(n)" is defined as "n is the Godel number of a formula that is provable in U", which means "there is a sequence of formulas, each of which is an axiom or derivable from the axioms, and n is the Godel number of the last formula in that sequence". But "is an axiom" is defined in a way that presupposes having defined "provable_U", thus circular.

So I don't see how it can be a well-defined, let alone recursive axiomatization, of U, though it is a recursive axiomatization of another theory U'.

It was claimed that there are capable philosophers of mathematics who are unfamiliar with basic logical notation. Maybe there are, but I don't know of any other than philosophers prior to the widespread use of notation in mathematical logic. To responsibly philosophize in modern context about subjects such as incompleteness does require understanding at least the basic technical material. We can outline certain notions with minimal notation, but at a certain point, if we wish not to oversimplify at the cost of being misinformational, we need to use some symbolisms. (For the most excellent discussion for the layman of incompleteness, refer to Franzen's 'Godel's Theorems', which is a beautifully written and truly insightful presentation.)

I am rusty in the subject of incompleteness, so my remarks might require corrections, but here is a sketch of some of the terminology and notions. To truly understand these notions, starting from the beginning, I recommend Kalish, Montague and Mar's 'Logic: Techniques Of Formal Reasoning' as an introduction to symbolic logic, then onto Enderton's 'Set Theory' for needed set theoretical context, then Enderton's 'A Mathematical Introduction To Logic' for mathematical logic including incompleteness):

(Throughout, where certain results are actually due to Godel-Rosser, for simplicity I'll refer merely to Godel. Also, Godel himself did not refer to 'the standard model', as the notion of a formal model was formalized a few years later by Tarski, but it's easier to discuss Godel with the notion, as the notion can reasonably be said to be implicit with Godel.)

PA is a theory in first order logic with identity, with the signature:

0 - constant, intuitively 'zero'
S - 1-place operation, intuitively 'successor'
+ - 2-place operation, intuitively 'addition'
* - 2-place operation, intutively 'multiplication'

The axioms of PA:

For all n and k, and for all formulas F:

0 not= Sn
i.e. 0 is not a successor

if Sn = Sk, then n = k
i.e. the successor function is 1-1

n+0 = n

n+Sk = S(n+k)

n*0 = 0

n*Sk = (n*k)+n

if (F(0) and for all n, F(n) then F(Sn)), then for all n, F(n)
i.e. (thinking of F as expressing a property) if F is true of 0 and whenever F is true of a natural number then it is true of that number's successor, then F is true of all natural numbers.

PA is a formal theory (i.e., a recursively axiomatized theory) in the sense that it is computable whether any given formula is or is not an axiom, and it is computable whether any given sequence of formulas is or is not a proof.

/

Let w = the set of natural numbers.

The standard interpretation of PA is the structure:

<w, zero, successor_on_w, addition_on_w, multiplication_on_w>

/

(Throughout, by 'symbol', 'sequences of symbols', 'sequence of sequences of symbols' I mean of PA.)

A formula is a certain kind of sequence of symbols. A sentence is a certain kind of formula. A proof is a certain kind of sequences of formulas. A theorem is a sentence that is the last entry in a proof. A theory is a set of theorems. So PA is the set of theorems derivable from the axioms of PA.

A theory is consistent if and only there is no sentence in the theory such that both it and its negation are members of the theory, which is to say that there is no sentence such that both it and its negation are theorems of the theory.

A theory is complete if and only if for any sentence, either it or its negation is a theorem.

Note that any inconsistent theory is complete, since an inconsistency proves every sentence.

The incompleteness theorem pertains to many theories, but in this thread we are looking in particular at PA, so the incompleteness theorem for PA is: If PA is consistent then PA is not complete. (To simplify discussion, throughout I will assume that PA is consistent. With the assumption that PA is consistent, the incompleteness theorem is: PA is incomplete.)

Godel showed 1-1 functions having pairwise disjoint ranges:

g from the set of symbols into w
g' from the set of sequences of symbols into w
g'' from the set of sequences of sequences of symbols into w

For any symbol, sequence of symbols, or sequence of sequences of symbols it is computable what its value is per the functions, and for any natural number it is computable what its symbol, sequence of symbols, or sequence of symbols is, if any, per the inverses of the functions.

For any s that is a symbol, sequence of symbols, or sequence of sequences of symbols, we may refer to #(s), which is the value of s per the pertinent function. We call #(s) "the Godel number of s".

/

(Throughout, by 'sentence', 'proof', and 'theorem' I mean of PA.)

Godel then showed how such predicates as 'is a sentence', 'is a proof', and 'provable' can be defined in the language of PA.

Then Godel showed a particular sentence, call it 'G', such that G <-> not-provable(#(G)), that is, #(G) is not the Godel number of a theorem of PA, which is to say that G is not a theorem. Moreover, the negation of G is not a theorem, since it is a theorem if and only if it is not a theorem, which is impossible. So there is a sentence such that neither it nor its negation is a theorem, which is to say that PA is incomplete.

/

But to PA we can add G itself as an axiom. Call this theory PA'. And PA' is consistent, but there is a sentence G' of PA' such that neither G' nor its negation are theorems of PA'. Ad infinitum.

angry people — SkyLeach

I'm not angry about you.

you're literally talking about writing style — SkyLeach

No, that's the point, I'm not. I'm talking about the very concept, as it was mentioned to you by even another poster. Moreover, even if it were merely (which it is not) notation, then still it is hard to imagine anyone has studied linear algebra without having seen this utterly ubiquitous notation.

Anyway, now that I've given you an explicit explanation, you may understand the difference between a set taken in and of itself and an ordered tuple, and you now have the common notation you may use going forward to contribute to communicating clearly about mathematics with people who have actually studied it and know about linear algebra..

AFAIK one doesn't use braces for subsets — SkyLeach

This distinction between, on the one hand, a set in and of itself without specifying an order and, on the other hand, an ordered tuple is crucially basic to mathematics, especially linear algebra.

Curly braces are used for sets in general. Either by set abstraction such as

{x | x is a natural number less than 4}

or for specifying the members individually such as

{0 1 2 3 }

where the order in which the members are listed, or redundancies are irrelevant, so:

{0 1 2 3} = {3 1 0 2} = {3 3 1 0 2} etc.

Angle brackets [alternatively, parenthesis] are used for order tuples such as

<0 1 2 3 > [alternatively (0 1 2 3)]

and order and redundancy do matter, so none of the below are equal to one another:

<0 1 2 3>
<3 1 0 2>
<3 3 1 0 2>
etc.

/

This is not just a matter of notation, but is a crucial concept, especially in linear algebra. I don't know why someone would be posting such bold claims as yours about mathematics and linear algebra while not even knowing that there is a distinction between merely a set and an ordered tuple.

Saying that linear algebra is the foundation of mathematics while not knowing the basic notion of an ordered tuple is like saying benzene rings are the foundation of chemistry while not knowing what an atom is.

(good) analogies. — Agent Smith

It's a ridiculous analogy.

How can something that can't be counted be mathematical? Can consciousness be counted? — Agent Smith

Are you arguing this way?:

The set of real numbers can't be counted.
Consciousness can't be counted.
Consciousness is not mathematical.
Therefore, the set of real numbers is not mathematical.

Oh my god! You discovered the hidden truth that there is a rupture in mathematics! Division is not closed in the integers! A discovery as shocking as that Soylent Green is people! And there is not just your example, but thousands of them! Millions of them! Maybe even infinitely many of them! And this contagion is not confined just to mathematics but it affects even the entire garment industry!

You didn't ask — SkyLeach

Of course I did. It's on record as a post.

Meanwhile, your characterization of mathematicians, as a generalization, is unsupported by any data whatsoever.

Counting lies at the heart of mathematics. An uncountable object (e.g. the set of reals), therefore, must be, in some way, nonmathematical, oui? — Agent Smith

Non.

Function, as in what the purpose reason has been evolved to fulfill. — Garrett Travers

Okay. Still, the Objectivist ethical thesis is not supported by valid argument.

Then there was no reason to waste our time on it. — Garrett Travers

My point did not depend on equality. I explained my point long ago. And I have pointed out that you chronically skip the telling points in my original argument and the further explanations in response to you.

I'll pick this up tomorrow. — Garrett Travers

So I guess your promise to another poster that you wouldn't bother him with your Objectivism if he read ITOE and found it wanting doesn't apply also to me.

/

You said you recently aced a course in logic. What was the textbook?

Because it's two in the morning — Garrett Travers

It's been two in the morning for 16 hours for you since your second post.

nonsense — Garrett Travers

Yours.

Yes, ethical conceptualizations are self-benefitting actions. Self-oppositional actions are not ethical — Garrett Travers

Moreover, that should be bidirectional. With both directions, my original formulation is tantamount to yours.

Whatever 'relate in function means', my not mentioning it is not an assertion of an equality.

No, it's a biconditional. — Garrett Travers

Good. That's what I said it is. I don't know why you said it isn't then changed your mind.

Another abysmally vague term.

And I made no claim of sameness of caliber nor containment in the "same ballpark" or anything like that.
— TonesInDeepFreeze

Then they wouldn't have been brought up together. — Garrett Travers

That is absurd. Bringing up two items for consideration together is not in and of itself equating them.

If and only if it is a benefit to one's own survival? — Garrett Travers

I don't see the point in quibbling whether we say "to benefit one's own survival" or "selfish".

Except, again, Objectivism says not just survival, but surviving to enjoy the exercise of one's rational values (which, is pretty much the Objectivist notion of selfishness).

And you just said it is not in 'if and only if ' form, yet you present your own version in 'if and only if' form.

It isn't an if and only if equation. — Garrett Travers

'if and only if' is not an equation; it is an equivalence.

You don't think that Objectivism holds that if an act is ethical then it is selfish, and that if an act is selfish then it is ethical?

And even if it were only one direction of the arrow, no matter which direction, it doesn't follow by valid logic in the Objectivist arguments.

That does not entail that an act is ethical if and only if it is selfish.
— TonesInDeepFreeze

That's not Objectivism's argument: — Garrett Travers

I didn't say it is an argument. it's the conclusion of an argument

You don't think that Objectivism claims that an act is ethical if and only if is selfish?

same caliber — Garrett Travers

Another abysmally vague term.

And I made no claim of sameness of caliber nor containment in the "same ballpark" or anything like that.

I read your words — Garrett Travers

You read without understanding simple English. Or you understood, but chose to skip the most telling points. Then eventually to resort to putting words in my mouth.

Individual humans are the source of ethics, ethics are standardized by that individual. — Garrett Travers

That does not entail that an act is ethical if and only if it is selfish.

I don't know what you thought you were getting across, — Garrett Travers

I said exactly what my argument is.

If you mean to say that you are looking for an is/aught distinction — Garrett Travers

I do not.

it isn't at all difficult to see why such a standard would be made — Garrett Travers

Saying "it is not at all difficult to see" is not a logical argument.

I'm pointing out that your argument against reason — Garrett Travers

You claim that I asserted an equality. I did not.

And that is not "equating" one with the other.
— TonesInDeepFreeze

It is — Garrett Travers

I don't mistake that breathing and reason are not the same thing. I don't mistake that breathing and reason do not share all properties. I don't mistake that even the properties they do share are not necessarily held to the same extent. I don't claim that breathing is more important or less important than reason.

If you persist in insisting that I have claimed an equality, then you will be persisting in lying about me.

reason is how the human navigates the world. — Garrett Travers

Even if we accept that in some general sense as true, it does not entail that an act is ethical if and only if is selfish.

an attempt to reduce the role reason plays in human survival, which supercedes that of breathing. — Garrett Travers

Now you're lying by putting words in my mouth.

Breathing is an action directed to survival.
— TonesInDeepFreeze

What kind of survival? Can babies live by just being left alone to breathe? — Garrett Travers

You are doing it again. At this point it is trolling.

Breathing is directed to survival. Breathing is necessary for survival. Breathing is not sufficient for survival.

Reason is directed to survival. Reason is necessary for survival. Reason is not sufficient for survival.

And that is not "equating" one with the other.

"You cannot do math without reason."

"Yeah but you cannot do reason without breathing."

Okay. Not the same ballpark of stuff. — Garrett Travers

'ballpark' is abysmally vague.

And my argument doesn't require that one is "in the ballpark" of the other. I have explained that; you skip,

You are equating autonomic functions with the cognition by which humans standardize all of their behavior. — Garrett Travers

That is false. I asserted no such equation. I said that both are necessary for survival. I did not in any way claim that they are equal (in a sense of identity) nor equal in importance (I don't even know how that would be measured) nor equal in primacy ('primacy" not even defined) nor equivalent in any way. They share the property of being necessary; that is not a claim of "equality".

And, again, the Objectivist argument is that it is only humans that use reason as the primary agency for survival.
— TonesInDeepFreeze

Yes, it is the human's primary agency for survival. — Garrett Travers

'primary agency' requires definition.

And, you skipped my main point, again. Even if reason is primary, that fact does not entail that an act is ethical if and only if it is selfish.

So putting aside "ensuring", breathing definitely is active and necessary for survival.
— TonesInDeepFreeze

Sure, it just isn't in the realm of guided behavior or thought devised for survival. — Garrett Travers

That is so specious.

I said that breathing is necessary and it's not reason. You reply, essentially yeah but its's not reason (guided thought).

Seek, pursue, attempt, or direct. — Garrett Travers

Breathing is an action directed to survival.

act in the world to ensure survival. — Garrett Travers

I addressed 'ensure'.

And breathing is an action. It is necessary for survival, but does not ensure it. Using reason is an action. It is necessary for survival but it does not ensure it.

I stated clearly that there's a difference between what happens when you respond in reflex, which informs later behavior, and what we know of the bug as far as the same phenomenon. — Garrett Travers

You said that there are things we don't know about creatures, but for many posts you claimed that even the least response to stimuli involves reason (even, if I recall (I'm not going back now to find the quotes), that it is an instance of reason). I even pointed out that knee jerk is not exercise of reason, and you persisted to claim that it is (the point about knee jerk not being a survival mean is not relevant to the mere question of what is and is not reason).

Breathing is not exercise of reason, and not plausibly at any "level". So reason is not the only thing necessary for survival, which is to take exception to your original description in your earlier flubbed argument.

Now you move the goalposts to what is "in the same realm", while that is quite vague terminology, and my argument doesn't rely on realm anyway.

me not understanding that that's literally all you were saying. — Garrett Travers

Clearly, it is not all that I am saying. It was part of an argument I layed out very clearly, and with any rebuttals needed to advance it against your specious replies, and even as your specious replies chronically skipped key points I made.