You regularly and willfully spread ignorant disinformation about mathematics throughout many threads, over the course of at least several months, even repeating yourself after you've been given detailed explanations why you are incorrect. It's disgusting.

Agent Smith is poking at us. He is much more intelligent than he seems.

It's disgusting. — TonesInDeepFreeze

Gives TonesInDeepFreeze 8 mg of Zofran.

Zofran — Agent Smith

From your personal stash. Side effects include confusion and disorientation. Explains a lot.

From your personal stash. Side effects include confusion and disorientation. Explains a lot. — TonesInDeepFreeze

Perhaps! :snicker:

Agent Smith is poking at us. He is much more intelligent than he seems. — jgill

:chin:

Agent Smith is poking at us. He is much more intelligent than he seems. — jgill

Gives TonesInDeepFreeze 8 mg of Zofran. — Agent Smith

TIDF can be very entertaining that way, and also quite enlightening. But eventually it gets boring when TDIF refuses to divulge the secrets of the mathemagician's smoke and mirrors.

In a sense then ∞∞ in science has a job description similar to contradictions - separates the possible from the impossible. — Agent Smith

This is actually a very good point. Every time infinity is employed in the application of mathematics, it's like employing a contradiction. This becomes very clear in an analysis of the common mathematician's claim to have resolved Zeno's paradoxes.

:ok:

Which means $\infty$ is impossible, squaring with Aristotle's decision to make the distinction potential vs. actual (infinity).

From special relativity, the Lorentz factor is unbounded as v approaches c. — jgill

If all the energy of the universe were somehow marshalled to accelerate a single particle, what would its Lorentz factor be? I have no clue, but I suspect it wouldn't come near 23^2017^3138934, let alone a real monster like 10^10^10^10

Well, from my study of astronomy and mathematics, maninly the ingrendients added by such thinkers as Hawking and Cantor, I can conclude that there are infinitities within infinites The max number seems to cause a break in physical laws that hold the universe outsside of the infinite together..This infinities in infinities are indicative in empirical findings of black-holes (something that fastinated me in my younger years). These have been observed to exist just recently. Once forces reach their max, the system collapses on itself. What happens exactly (completly) at such points is still an unresolved mystery of astronomers and mathematicians alike. I have my own postulates but that is irrelavant here.

What is the size of the set of possible states of the universe? I suspect this is the true nmax. Would this be a number best expressed as x^y^z?

I think 24 is enough for most things.

when TDIF refuses to divulge the secrets of the mathemagician's smoke and mirrors. — Metaphysician Undercover

There is no magic. Very much to the contrary. At a bare minimum, it is algorithmically verifiable whether a given formal expression is well formed and then whether a given sequence of formulas is a formal proof. That is a courtesy given by formal logic that is not hinted at in various handwavings and posturings by cranks as often found in a forum such as this. And I have given extensive explanation of many of the formulations I have mentioned.

Every time infinity is employed in the application of mathematics, it's like employing a contradiction. — Metaphysician Undercover

Yet no one who says things like the above has ever demonstrated that Zermelo set theoretic infinitistic mathematics implies a contradiction.

This becomes very clear in an analysis of the common mathematician's claim to have resolved Zeno's paradoxes. — Metaphysician Undercover

Zeno's paradox is not a formal mathematical problem. Saying that calculus provides a problem solving tool in which paradox does not occur does not imply a contradiction.

Yet, again, we remind that a contradiction is a statement of the form "P and not-P". No such statement has ever been derived from Zermelo set theoretic infinitistic mathematics, no matter that, perpetually, cranks groundlessly and ignorantly claim otherwise.

.

The statement that there exists sets that are infinite is not a logical impossibility.

The idea of allowing 'there exist potentially infinite sets' but not 'there exits infinite sets' is fine as a motivation for an alternative mathematics. But, if one cares about mathematics being formal (in the sense that it is utterly objective by algorithmic checking whether a sequence of formulas is indeed a proof from axioms), then the notion of 'potential infinity' requires formal definition and axioms to generate the desired theorems about it.

Saying, "I don't like the notion of infinity so I'll use potential infinity instead" but without even hinting at how that alternative would be formulated is no better than saying "I don't like that human life requires breathing oxygen so we should breathe hydrogen instead" but not giving a hint as to what technology would allow hydrogen to do the job of oxygen.

There is no magic. Very much to the contrary. At a bare minimum, it is algorithmically verifiable whether a given formal expression is well formed and then whether a given sequence of formulas is a formal proof. That is a courtesy given by formal logic that is not hinted at in various handwavings and posturings by cranks as often found in a forum such as this. And I have given extensive explanation of many of the formulations I have mentioned. — TonesInDeepFreeze

Whether an expression is well formed or not is irrelevant to whether it is self-contradictory, because to determine contradiction we must analyze the meaning, and this is the content, not the form.

Yet no one who says things like the above has ever demonstrated that Zermelo set theoretic infinitistic mathematics implies a contradiction. — TonesInDeepFreeze

Have you forgotten the conversations we've had earlier? The empty set for instance, involves contradiction.

Yet, again, we remind that a contradiction is a statement of the form "P and not-P" — TonesInDeepFreeze

Oh, now I remember, you have a very odd notion of what constitutes contradiction, and this is how you insist that there is no contradiction even after contradiction is demonstrated to you. If the statement doesn't explicitly say "P and not-P", then there is no contradiction in your interpretation, regardless of what the statement means.

The law of noncontradiction states that the same object cannot both have and not have, the same property, at the same time, in the same respect. So consider the empty set for example. For simplicity, let's say that a set is a collection of objects. Therefore a set necessarily has objects. The empty set has no objects. Therefore "empty set is self-contradicting. The empty set is said to have objects (necessary to being a "set", or collection of objects) and also to not have objects (necessary to being empty), at the same time, and in the same respect

Whether an expression is well formed or not is irrelevant to whether it is self-contradictory, because to determine contradiction we must analyze the meaning, and this is the content, not the form. — Metaphysician Undercover

You are entirely ignorant of what contradiction is in mathematics.

Moreover, even if contradiction were, in some sense, couched semantically, then no contradiction, even in some sense of a semantic evaluation, has been shown from ZFC.

Moreover, if an expression is not grammatical, then it does not admit of semantic evaluation.

The empty set for instance, involves contradiction. — Metaphysician Undercover

"Involves contradiction" has not been given meaning by you. Either the theorem that there exists an empty set implies a contradiction or it does not. No contradiction has been shown to be derived from the theorem that there exists an empty set.

If you mean that the notion of 'set' is not compatible with a set being empty, then that just entails that your conception differs from a different conception in which there is an empty set.

Moreover, the phrase "the empty set" is not in the formal theory. Rather, there is a theorem:

E!xAy ~yex

and definition:

x=0 <-> Ay ~yex.

Moreover, even if you persisted to object to mathematicians using the informal locution 'the empty set', then mathematicians could say, "Okay, we won't say 'empty set' anymore. Instead we talk about sets and one particular object, whether it is a set or not, such that that object is the only urelement (an object that has no members), and then all these things - the sets and the urelement - are called 'zets'. So there is an empty zet." That would not alter the mathematics of set theory one bit, especially formally, and even informally except that the mouth pronounces a 'z' instead of an 's' for that one word.

So you are terribly ignorant and self-misguided in every aspect of this matter.

you have a very odd notion of what constitutes contradiction — Metaphysician Undercover

No, I have the standard logical and mathematical notion.

The law of noncontradiction states that the same object cannot both have and not have, the same property, at the same time — Metaphysician Undercover

That is one informal formulation. It is equivalent though to the standard formulation. That is:

~Ex(Fx & ~Fx)

is equivalent to

~(P & ~P).

let's say that a set is a collection of objects — Metaphysician Undercover

First, 'set' is not a primitive of set theory. An actual definition can be:

x is a class <-> (x=0 or Ey yex)

x is a proper class <-> (x is a class & ~Ey xey)

x is a urelement <-> (~x=0 & ~Ey yex)

x is a set <-> (x is a class & ~x is a proper class)

Second, even informally, you mention a certain definition of 'set'. Mathematicians are not then obliged to refrain from having an understanding in which "collection of objects" does not preclude that it is an empty collection of objects, notwithstanding that that seems odd to people who have not studied mathematics, and so more explicitly we say, "a set is a collection, possibly empty, of objects". You are merely arrogating by fiat that your own notion and definition must the only one used by anyone else lest people with other notions and definitions are wrong. That is an intellectual error: not recognizing that definitions are provisional upon agreement of the discussants and that one is allowed to use different definitions in different contexts among different discussants. It's like someone saying "a baseball is only one such that is used in major league baseball" and not granting that someone in a different context may say, "By 'baseball' I include also balls such as used in softball". It is intellectually obnoxious not to allow that. And it is one in the deck of calling cards of cranks.

You are entirely ignorant of what contradiction is in mathematics. — TonesInDeepFreeze

OK, Mathematics is allowed its own special definition of "contradiction", so that statements which would qualify as contradictory in a rational field of discipline do not qualify as contradiction in mathematics.

Second, even informally, you mention a certain definition of 'set'. Mathematicians are not then obliged to refrain from having an understanding in which "collection of objects" does not preclude that it is an empty collection of objects, notwithstanding that that seems odd to people who have not studied mathematics, and so more explicitly we say, "a set is a collection, possibly empty, of objects". You are merely arrogating by fiat that your own notion and definition must the only one used by anyone else lest people with other notions and definitions are wrong. That is an intellectual error: not recognizing that definitions are provisional upon agreement of the discussants and that one is allowed to use different definitions in different contexts among different discussants. It's like someone saying "a baseball is only one such that is used in major league baseball" and not granting that someone in a different context may say, "By 'baseball' I include also balls such as used in softball". It is intellectually obnoxious not to allow that. And it is one in the deck of calling cards of cranks. — TonesInDeepFreeze

When you explain to me how a set which has no objects also has a collection of objects, and this is not contradictory, then I'll start to listen to you.

So please explain to me, your understanding of "collection of objects" in which there is no objects. As far as I can tell, either you have a collection of objects, or you have no objects, but to have both is clearly contradictory. What if you had one object? It is neither a collection of objects nor is it no objects. Do you agree? Or do you just abandon rationality for the sake of mathematics?

Mathematics is allowed its own special definition of "contradiction", — Metaphysician Undercover

It is a formal definition. But it still captures the ordinary sense of "To claim a contradiction is to claim a statement and its negation." For example, in ordinary conversation we may say, "'Mike is a car mechanic and Mike is not a car mechanic' is a contradiction."

explain to me how a set which has no objects also has a collection of objects — Metaphysician Undercover

You switched form "is a collection of objects" to "has a collection of objects".

I said nothing about "has a collection of objects". Rather, I said

"a set is a collection, possibly empty, of objects" — TonesInDeepFreeze

What if you had one object? — Metaphysician Undercover

An object that has no members is either the empty set or an urelement. And of course, an object that has in it only one object is a non-empty set.

So my explanation stands and all the rest of my remarks demolishing your ignorant and self-misleading remarks stand.

You're, how shall I say this?, forgetting the role of intuition in math. Vide Henri Poincaré.

Empty generalization and bluster.

I quite understand that human thinking, including about mathematics, involves intuition. Indeed I'm interested in the relation between formal theories and intuitions. And I know vastly more about the school of intuitionism compared with your lack of knowledge about it.

Empty generalization and bluster. — TonesInDeepFreeze

Henri Poincaré!

What is the size of the set of possible states of the universe? I suspect this is the true nmax. Would this be a number best expressed as x^y^z? — hypericin

I dunno! That's what I'm trying to find out.

Henri Poincaré! — Agent Smith

Well that truly settles the question!

Well that truly settles the question! — TonesInDeepFreeze

:ok:

And your emoticon doubly seals it! Who could ever defeat an emoticon?

And your emoticon doubly seals it! Who could ever defeat an emoticon? — TonesInDeepFreeze

You must free yourself (of mathematics).

And I know vastly more about the school of intuitionism compared with your lack of knowledge about it. — TonesInDeepFreeze

:ok: Please edify me then. How does intuition work in math? How is it related to so-called mathematical/logical rigor? Talking to you is like conversing with a computer. DOES NOT COMPUTE! DOES NOT COMPUTE! From start to finish, that's all you say! I should call tech support! :snicker:

Is there a finite number (Nmax) such that no calculations ever in physics will exceed that number? — Agent Smith

Calculations in physics. The Lorentz factor is unbounded.

The Lorentz factor is unbounded. — jgill

I'll havta take your word for it.

Suppose there was an upper bound to the Lorentz factor, $\gamma \lt M$.


Then the free variable v would also be bounded below c, which, in theory and thus computation, it is not:


$0\lt v \lt c \sqrt{1-\frac{1}{{{M}^{2}}}} \lt c$

How does intuition work in math? — Agent Smith

As it does in everyday life. But that is not the meaning of intuitionism in the philosophy of math.