Good. You are at your most eloquent with emojis. — TonesInDeepFreeze

Thanks for the compliment. Good day.

Could you give an example of a non-logical axiom — Tate

Each axiom of group theory is a non-logical axiom.

what makes it non-logical? — Tate

A non-logical principle is one that is not true in at least one model. — TonesInDeepFreeze

1. Classical logic has to use Occam's broom (sweep paradoxes under the rug) otherwise, via ex falso quodlibet, concede that classical logic is trivial.

2. We're using some version of paraconsistent logic and we're not aware of it. — Agent Smith

Classical binary Logic is best used for problems that can be precisely defined with integer numerical values. But human contradictions are seldom concisely defined; instead loosely sketched with inexplicit subjective truth-values.

A formalized version of "paraconsistent logic" (logic of paradox) is the Fuzzy Logic that is used in computer science for complex puzzles that are hard to define numerically, such as human beliefs & intuitions. It is especially useful in Artificial Intelligence and Evolutionary Programming.

We're not aware of our sloppy logic because it is intuitive, so we don't normally examine it with classical rules in mind. That's why divisive emotional issues, such as Abortion & Racism tend to polarize people. And can only be resolved, to some degree, with critical (rational) thinking : to discover the inconsistencies in our beliefs. :cool:


Paraconsistent Logic :
A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic which reject the principle of explosion.
https://en.wikipedia.org/wiki/Paraconsistent_logic

Most if not all thinkers are under the impression that they're using classical logic - they don't take too kindly to contradictions. — Agent Smith

Actually, most thinkers have an ego-boosting impression of their own reasoning abilities. We find it easier to see the contradictions in other people's ideas than in our own thoughts. Intuition always seems true, even when it aint.

A great many people think they are thinking when they are merely rearranging their prejudices.
— William James.

A formalized version of "paraconsistent logic" (logic of paradox) is the Fuzzy Logic — Gnomon

If I'm not mistaken, there is work in combining formal paraconsistent logic with formal fuzzy logic. But fuzzy logic itself is not a formalization of paraconsistent logic.

The diagonal argument is constructive and intuitionistically valid — TonesInDeepFreeze

The argument only goes to show that the continuum cannot be broken up in points. Leading to confused notions of infinitesimals or differentials.

BTW, brother Smith is not ad confused as you suggest. His contribution is a welcome addition to the body of philosophy. He makes the face of philosophy laugh.

The diagonal argument is constructive and intuitionistically valid
— TonesInDeepFreeze

The argument only goes to show that the continuum cannot be broken up in points. — Hillary

The proof shows that there is no enumeration of the set of real numbers. As formalized, the proof uses only first order logic from the axioms of Z set theory, and it could even be Z without the axiom of infinity by couching the proof with an hypothesis, rather than a given, that there exists an infinite set. Moreover, as I mentioned, the proof can even be couched in constructive mathematics with a notion of a "potential infinity" as a computational process without an upper bound of actions.

And the proof shows no such thing that the set of real numbers is not made of points. The continuum is <R less-than-on_R>. By definition, every real number is a point. It's a definition and does not require proof, and the diagonal argument does not contradict it.

Leading to confused notions of infinitesimals or differentials. — Hillary

Ordinary real analysis does not use infinitesimals. And the non-enumerability of the set of real numbers does not lead to use of infinitesimals.

Smith is not ad confused as you suggest — Hillary

I demonstrated exactly the manner in which he is confused and misinformational.

>. By definition, every real number is a point — TonesInDeepFreeze

But how you glue two points together. How can you construct a continuum with points as building blocks?

You can throw as many points in the bag as you like. It's never filled.

glue two points together — Hillary

throw as many points in the bag — Hillary

Those are your personal, impressionistic locutions. Real analysis doesn't have such terminology.

How can you construct a continuum with points as building blocks? — Hillary

The real continuum is constructed in formal axiomatic set theory. We prove that there exists a set and an ordering on that set (which are unique up to isomorphism) such that the ordering has the completeness property.

Your question suggests that you are not familiar with the basics of the subject.

Your question suggests that you are not familiar with the basics of the subject. — TonesInDeepFreeze

How many points do I have to throw in the bag to fill it?

A non-logical principle is one that is not true in at least one model. — TonesInDeepFreeze

Did Cantor's original set theory have non-logical axioms?

Time to walk away, @TonesInDeepFreeze.

On an unrelated, or at least only semi-related subject, does the fact that light has both a wave and particle nature constitute a valid example of a real-life, concrete paradox, which I just denied the existence of? — T Clark

particle-wave duality is only inconsistent in so far, that it doesn't mesh well with our typical view of the universe. it's however perfectly consistent, experiments will deliver consistently similar results. particle-wave duality is just a name to at least somewhat visualise what is happening in the equations of quantum mechanics

You can throw as many points in the bag as you like. It's never filled. — Hillary

well it will be filled after a certain infinite time, based on the size of each point.

Those are your personal, impressionistic locutions. Real analysis doesn't have such terminology. — TonesInDeepFreeze

You got some way of putting it, I have to admit! "Impressionistic locutions. Sounds like a new art form.

Points come in sizes?

But how [do] you glue two points together — Hillary

Gluing schemes in mathematics

I thought I remembered something about gluing in math, so I looked it up. Maybe not precisely what you ask, but interesting nevertheless.

Points come in sizes? — Hillary

In the real nos, no. But a "point" in a vector space or simply space in math frequently is a function of some kind, like a contour in C, and these might have "sizes".

I can imagine closing an infinitely small hole with a point, like adding one point closes an open interval. But throwing points in a bag...?

particle-wave duality is only inconsistent in so far, that it doesn't mesh well with our typical view of the universe. it's however perfectly consistent, experiments will deliver consistently similar results. particle-wave duality is just a name to at least somewhat visualise what is happening in the equations of quantum mechanics — TieableCookie

That was my point. I was trying to highlight that the only real paradoxes are logical or linguistic, which are trivial. There are no paradoxes in the real world, only stuff we don't understand or that surprises us.

How many points do I have to throw in the bag to fill it? — Hillary

'throw in', 'bag', and 'fill' (in your context) are not mathematical terms, so I can't give you a mathematical answer to your question.

However, the mathematical question "how many 3D-points are in a non-empty volume?" does have the mathematical answer: the cardinality of the set of real numbers.

Did Cantor's original set theory have non-logical axioms? — Tate

Cantor didn't have axioms. But of course he did use non-logical principles even if not formalized as axioms.

Except for the pure predicate calculus itself, any mathematical theory (such as formalized in predicate logic) has non-logical axioms.

Cantor didn't have axioms. But of course he did use non-logical principles even if not formalized as axioms. — TonesInDeepFreeze

Does this mean you're sort of stretching the idea of non-logical axioms to address the problems associated with naive set theory?

It appears that axioms were created specifically to block the path to Russell's Paradox. There isn't any logical basis for accepting that blockade.

Does this mean you're sort of stretching the idea of non-logical axioms to address the problems associated with naive set theory? — Tate

Sure, where the set theory is not formalized with axioms, we can at least point out that the pre-formal principles it uses are non-logical, at least in the sense that they would be non-logical axioms if they were formalized.

It appears that axioms were created specifically to block the path to Russell's Paradox. — Tate

It is common for people to put it that way, but it could be misleading to put that way.

Adding axioms cannot block the derivation of statements that are already derivable without the added axioms (i.e. the logic is monotonic). What axiomatic set theory does to avoid Russell's paradox is not add axioms but to refrain from adding the axiom schema of unrestricted comprehension that results in Russell's paradox. Then other axioms are added that permit derivation of the desired mathematical theorems.

Specifically, Zermelo refrained from adding the schema of unrestricted comprehension, so that schema is not available to use to derive Russell's paradox. Then Zermelo does add the schema of separation instead of unrestricted comprehension and the other axioms of Z set theory.

And Fraenkel did the same except he adds the schema of replacement (which is strong than separation but weaker than unrestricted comprehension) rather than the schema of separation. (I think the schema of replacement is important mainly for deriving transfinite recursion, which can't be derived in mere Zermelo set theory.)

the pre-formal principles it uses are non-logical, — TonesInDeepFreeze

In the sense that these principles are untrue in some models? That doesn't make any sense to me. How can a principle be false?

In the sense that these principles are untrue in some models? That doesn't make any sense to me. How can a principle be false? — Tate

It would be false in some models if it were formalized as a first order sentence, or, for a schema, it would have false instances if the schema were formalized.

The important point for the prior discussion here is that Cantor did not confine himself to merely logical principles. The Cantorian paradoxes are not a failing of logic.

It would be false in some models if it were formalized as a first order sentence, — TonesInDeepFreeze

What would that sentence be?

The Cantorian paradoxes are not a failing of logic. — TonesInDeepFreeze

I agree.

What would that sentence be? — Tate

The schema of unrestricted comprehension specifies an infinite number of axioms:

If F is a formula and b is a variable that does not occur free in F, then all closures of

EbAy(yeb <-> F)

are axioms.

Every instance of that schema is a non-logical axiom. [Edit: I have to think more closely whether every instance is non-logical. But still, the point prevails that unrestricted comprehension has non-logical instances, and in particular an instance used to derive Russell's paradox is not just non-logical but it is also inconsistent.]

Moreover, the particular instance:

EbAy(yeb <-> ~yey) is not just false in some models but it is false in every model, as we know from Russell's paradox.

Note also that this is not specific to set theory. To derive Russell's paradox from unrestricted comprehension, we don't rely on anything specific about 'e', the membership relation. Rather for any 2-place predicate R whatsoever (whether 'R' stands for 'is a member or' or 'shaves' or 'is a parent of' ...) we derive a contradiction from:

EbAy(Ryb <-> ~Ryy)

What's the model where an instance is false?

See post above that I added to.

EbAy(yeb <-> ~yey) is false in not just some models but it is false in every model.