Mersenne's Miscelleny

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James Strom
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Re: Mersenne's Miscelleny

Post by James Strom »

Your idea can become a YouTube sensation! Just get a snake to chase its own tail around some circular object.

"And your chums weren't cheating; if you think of a googleplex and your friend names a googleplex plus one, he's thought of a bigger number."

But the person wouldn't write "a googleplex plus one" but rather "your number plus one." What if I had written the same thing? What would the number be then?

"It goes to the limit. And this limit exists, because the function 1/x is differentiable."

But isn't the limit at 0? Or is it just before it but not 0 itself? If the latter is the case then it is either:
1) The graph is finitely detailed and so the limit is a small but nonzero distance from 0 in which case infinity is huge but finite.
2) The graph is infinitely detailed so the limit is an infinitesimal distance from 0 in which case an infinity is necessary to define infinity. This would be a tautology.

When most mathematicians use an infinity they are really thinking of "A really big number." You could replace infinity in just about any equation with 2^25964951-1 and it would work just as well.
Everything hinges on whether infinity is a number or not. I think we may have a case of doublethink here. Infinity is treated as if it were a number for some purposes but not for others. If, as Peano has "proven", the number of natural numbers is infinite then infinity is a number. But when asked for details about this number it is then claimed that it is not a number. I put quotes on "proven" because I may not accept all his axioms.

"Every infinity is greater than any number, so it can't be a number"

It can be a subjective number. I could define infinity as whatever number you can come up with plus one. It will therefore be larger than any number yet still be a number. But, as usual, no subjective number can ever be precisely defined. As soon as we try to pin it down it changes. So it is not a number in the objective sense. This is what I think is meant when people imagine infinity. In the subjective world things can both be and not be simultaneously.
Cantor and others have tried tricking us into thinking that infinity can exist in an objective manner. This is done by dressing it up in formulae and "proofs" that look pretty good until they're looked upon closely. As soon as a contradiction or tautology is found they make it seem as if you only need to have a deeper understanding to get it. But, just like with Bigfoot, the actual proof is never acquired. It's always "just around the corner."
This is fine in the subjective world. That is how it operates. To know yourself is to change yourself so a final definitive answer becomes impossible. But mathematics and logic are supposed to be objective, and rightly so. Definitions are supposed to be exact so that any question can be resolved, at some point, with certainty.
So why would Cantor and others try deceiving us into thinking infinity is objective? My guess would be that it follows the pattern in the modern age of those who tear down our spiritual selves. Some scientists will make the nonsensical claim that now that evolution has been discovered that we can scientifically determine the meaning of life. Dawkins, for example. But this is absurd. Science can only tell you what is, not what should be.
If infinity is subjective then it is a part of our spiritual, mystical selves. If infinity is then treated as nothing more than a objective mathematical concept our souls die in some way. We lose purpose and meaning in life.
To prevent this is my motivation in trying to prove to you the non-objective nature of infinity. That doesn't make it non-objective, of course. But once you see it this way it becomes a lot harder to excuse the rationalizations that Cantor and others come up with for dispensing with ordinary logic and reason. Profundity is no defense.
Sorry for having such a strong opinion. I suppose infinity ought to be added to religion and politics as questionable topics for discussion. Or not.

On a lighter note you certainly have got me thinking. Last night I was pondering what one divided by zero was. It has always bothered me that it remains undefined. Math shouldn't have such things, especially something that doesn't involve anything more complex than a one, division operator and a zero.
So I tried working it backwards. Division is only multiplicaton in reverse. A/B=C can be rewound with C*B=A. So if x=1/0 then x*0=1. Frustrating. Everything multiplied by zero is zero. But that can't be right.
When two numbers are multiplied the product contains the same amount of information as was put into the operation. This assumes the commutative law of multiplication is not valid if you don't line up the units afterwards. For example, using 1s to represent single units of a quantity it can be shown that A*B<>B*A.

2*3=
11
11
11

3*2=
111
111
The product contains the information needed to be able to determine exactly how the operation occurred.
However, if the 1s are lined up and then compared then the commutative law is valid.
111111=111111
But multiplication by zero doesn't allow for this. Multiply 4, 8, 32 or any other number by 0 and you get:

Nothing. So information must be lost. This is not good.
But what if it were viewed this way:
3*0=
000
4*0=
0000
Sure, in both cases it adds up to zero. But we don't have to add them. 3*0<>4*0.
What if we stopped assuming zero is a natural number? Suppose a zero is just another form of an integer? Imagine viewing numbers as blocks of white squares on a field of black ones. There is no reason why we cannot view it as a negative image. Then the zeros can be multiplied, divided, etc. with each other following the same rules as ordinary integers.
(2*0)*(3*0)=
00
00
00
So 1*0=
0
This is equal to one in the zero universe. So 1*0=1 in that black=white world. So 1/0=1*0, 2/0=2*0, etc. Kind of makes sense if you look at a graph of reciprocals. 1/0.2=5, 1/0.1=10, 1/0=0, 1/-0.1=-10, 1/-0.2=-5, etc.
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Re: Mersenne's Miscelleny

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Your idea can become a YouTube sensation! Just get a snake to chase its own tail around some circular object.{
Training is in progress.
But the person wouldn't write "a googleplex plus one" but rather "your number plus one." What if I had written the same thing? What would the number be then?
Then neither you nor your chum have specified a number. You can't have recurrence if there isn't an occurence.
ME: "It goes to the limit. And this limit exists, because the function 1/x is differentiable."

But isn't the limit at 0? Or is it just before it but not 0 itself? If the latter is the case then it is either:
1) The graph is finitely detailed and so the limit is a small but nonzero distance from 0 in which case infinity is huge but finite.
2) The graph is infinitely detailed so the limit is an infinitesimal distance from 0 in which case an infinity is necessary to define infinity. This would be a tautology.
The limit of the function is not at zero, it's at infinity. It's the independent variable x that approaches zero.
1/x can be made as close to infinity as desired by making x sufficiently close to 0. Because x can equal zero, infinity exists. When x = 0, the function is undefined (which doesn't mean that 1/0 doesn't exist, only that there is no number which, multiplied by 0, gives 1; it doesn't belong to arithmtic on the reals).
When most mathematicians use an infinity they are really thinking of "A really big number."
Crross my heart, we don't. That's what engineers do.
You could replace infinity in just about any equation with 2^25964951-1 and it would work just as well.
Not from a mathematician's perspective, though again an ngineer (or, I suppose, a physicist) would be happy.
Everything hinges on whether infinity is a number or not.
Infinity is not a number.
I think we may have a case of doublethink here. Infinity is treated as if it were a number for some purposes but not for others. If, as Peano has "proven", the number of natural numbers is infinite then infinity is a number.
Not so. There are an infinite number of natural numbers, but no one natural number lies at infinity.
But when asked for details about this number it is then claimed that it is not a number. I put quotes on "proven" because I may not accept all his axioms.
Inconsistency is not doublethink, but there is a single truth it's possible to be incnsistent about. That truth is that infinities exist and are not numbers, but that one type of infinity can be distinguished from another by associating these with different number sets.
ME: "Every infinity is greater than any number, so it can't be a number"

It can be a subjective number. I could define infinity as whatever number you can come up with plus one. It will therefore be larger than any number yet still be a number.
Sorry James, but this just doesn't work as a definition of infinity. Add one to a finite number and it's still just a finite number, whether it's known or not.
But, as usual, no subjective number can ever be precisely defined. As soon as we try to pin it down it changes. So it is not a number in the objective sense. This is what I think is meant when people imagine infinity. In the subjective world things can both be and not be simultaneously.
Cantor and others have tried tricking us into thinking that infinity can exist in an objective manner. This is done by dressing it up in formulae and "proofs" that look pretty good until they're looked upon closely.
I promise you I have looked at them closely. They look pretty good because they're mathematiccal proofs. Mathematics goes beyond the manipulation of numbers. Cantor was not tricking us when he evolved the diagonal slash argument, he was bettering our understanding.
As soon as a contradiction or tautology is found they make it seem as if you only need to have a deeper understanding to get it. But, just like with Bigfoot, the actual proof is never acquired. It's always "just around the corner."
The proofs are there. I've quoted them. I haven't given a foot-shaped impression with the words "a proof was here but has now vanished", I've given the limit definition of infinity.
This is fine in the subjective world. That is how it operates. To know yourself is to change yourself so a final definitive answer becomes impossible. But mathematics and logic are supposed to be objective, and rightly so. Definitions are supposed to be exact so that any question can be resolved, at some point, with certainty.
So why would Cantor and others try deceiving us into thinking infinity is objective? My guess would be that it follows the pattern in the modern age of those who tear down our spiritual selves. Some scientists will make the nonsensical claim that now that evolution has been discovered that we can scientifically determine the meaning of life. Dawkins, for example. But this is absurd. Science can only tell you what is, not what should be.
Leave poor Cantor alone, he was no con man.
If infinity is subjective then it is a part of our spiritual, mystical selves. If infinity is then treated as nothing more than a objective mathematical concept our souls die in some way. We lose purpose and meaning in life.
To prevent this is my motivation in trying to prove to you the non-objective nature of infinity. That doesn't make it non-objective, of course. But once you see it this way it becomes a lot harder to excuse the rationalizations that Cantor and others come up with for dispensing with ordinary logic and reason. Profundity is no defense.
Sorry for having such a strong opinion. I suppose infinity ought to be added to religion and politics as questionable topics for discussion. Or not.
So far from dispensing with logic and reason, Cantor greatly extended it. Godel's work was profoundly spiritual, and used Cantor's arguments.
On a lighter note you certainly have got me thinking. Last night I was pondering what one divided by zero was. It has always bothered me that it remains undefined. Math shouldn't have such things, ...
Well, maths doesn't use the term "undefined" to mean, say "non existent" or "not understandible". It only means that it has an answer placed outside the set on which the operation takes place. Your number manipulation is fun, but the infinity will always be outside the numbers being operated on.

BTW, every thought of a career in number theory?
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Re: Mersenne's Miscelleny

Post by James Strom »

I lack the education for it. Besides, I'm too old to get the Field's medal. I just love logic, philosophy, mathematics and such.
When I was very young I had a fascination with large numbers and spent a lot of time working out things like "superpowers" (X^^Y).*
But infinity has always been a non-starter for me. It requires a suspension of logic that I just am not willing to do when thinking about math. But maybe I think differently from others. I even have a problem with Cantor's theorem. I'm a strong believer in the Law of the Conservation of Information. So how could there be more information in the subsets than in the set itself?

*By the way, I've figured out how to do fractional superpowers. As far as I know no one else has ever been able to do this. Hint: rewind.
A-B=C. C+B=A.
A/B=C. C*B=A.
SQR(A)=C. C^2=A.
" "
" "
This is the only clue I'll give. I want my medal.

I have to go to job training but I'll refute all your points as soon as I can.
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Re: Mersenne's Miscelleny

Post by Mersenne »

Bring it on!

Though I may have to open a new thread for the rest of the Miscellany...

In the mean time, have you read "A New Golden Age" by Rudy Rucker (1981)?
So how could there be more information in the subsets than in the set itself?
Cantor's doesn't demand this. Nor does Godel's use of the diagonal slash. A set contains more information han its subsets, but not all may be relevant to a global issue, or accessible to the whole.
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Re: Mersenne's Miscelleny

Post by James Strom »

How do you put those quotes in the grey boxes? I can't figure that out. I'm stuck doing this:
"And your chums weren't cheating; if you think of a googleplex and your friend names a googleplex plus one, he's thought of a bigger number."
"But the person wouldn't write "a googleplex plus one" but rather "your number plus one." What if I had written the same thing? What would the number be then?"
"Then neither you You can't have recurrence if there isn't an occurrence."

How do you reconcile "he's thought of a bigger number" with "nor your chum have specified a number." That seems, or is, flat out contradictory.

"1/x can be made as close to infinity as desired by making x sufficiently close to 0."

Only if the graph is infinitely detailed. As I've pointed out you can't use something with an infinity in it to define infinity or you've got a tautology.

"Because x can equal zero, infinity exists."

So infinity is equal to 1/0? It either:

1) Is not. Then it is not necessary for x to be equal to zero to prove infinity exists and so the quote above makes no sense.
or
2) It is. Then infinity is undefined. Not a good thing to conclude when trying to define infinity.

Out of time.
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Re: Mersenne's Miscelleny

Post by admin »

When you start your 'Reply', highlight the part of the text in the other person's text that you want to appear in a grey box, right click and in the drop-down box choose 'Copy'.

Where you want that piece of text to appear in your reply, right click at the place you want to Insert it; and in the drop down menu, click 'Paste'.

Highlight that text and then, from the line of boxes immediately above, click 'Quote'.
"But the person wouldn't write "a googleplex plus one" but rather "your number plus one." What if I had written the same thing? What would the number be then?"
You'll now see that text with a bracketed
at the start and at the end of it.

When you click 'Submit', that piece of text will show up in a grey box.

Admin.
"Don't let the illusions of your past or future rob you of the infiniteness of your present." [Unknown]
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Re: Mersenne's Miscelleny

Post by Mersenne »

Hi James,
"But the person wouldn't write "a googleplex plus one" but rather "your number plus one." What if I had written the same thing? What would the number be then?"

ME:"Then neither...You can't have recurrence if there isn't an occurrence."

How do you reconcile "he's thought of a bigger number" with "nor your chum have specified a number." That seems, or is, flat out contradictory.
If you've named a googleplex and your chum named a googleplex plus one, he's named the bigger number. If you've both thought "your number plus one" neither of you have named a number. No contradiction, they are different instances.
ME: "1/x can be made as close to infinity as desired by making x sufficiently close to 0."

Only if the graph is infinitely detailed. As I've pointed out you can't use something with an infinity in it to define infinity or you've got a tautology.
.

A tautology is something which is always true- we explored that with respect to definitions. And I've already demonstrated that any number must be defined in terms of itself, so why demand otherwise of infinity, which is not even a number?

Moreover, that the graph (or rather, the space) is sufficiently detailed is a consequence that there can be zero distance, not of infinity- or even of the infinitesimal!

AND this is just one infinity, the infinity of the reals. Your own chequerboard universe would still support the infinity of the natural numbers.
ME: "Because x can equal zero, infinity exists."

So infinity is equal to 1/0? It either:

1) Is not. Then it is not necessary for x to be equal to zero to prove infinity exists and so the quote above makes no sense.
or
2) It is. Then infinity is undefined. Not a good thing to conclude when trying to define infinity.

Out of time.
1) Infinity is not equal to 1/0, but 1/x has passed through infinity when x=0. Like it or not, the quote makes rigorous mathematical sense.

2) If infinity were undefined, it would still exist. I explained what "undefined" means in a maths context.

Pardon me, James, but you seem to be disregarding many of my earler replies. I must assert that infinity exists. Infinities exist.
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Re: Mersenne's Miscelleny

Post by James Strom »

Thanks for the info on quotes. Very helpful.

I'm sorry if you think I'm disregarding your earlier replies. I'm trying to be thorough by covering every base. If, for instance, you say that infinity is not 1/0 I will still assume that you think it is a possibility because a later remark seems to indicate that you think it might. Basically, I'm trying very hard to make sure there are no inconsistencies in your arguments.

I think we'll get nowhere if we engage in semantics. If we're arguing over what the definition of 'undefined' is then things have gotten ridiculous.
I'll show you how I know you're wrong instead. Visualize the graph. Imagine that you can increase the magnification of your view until you eventually reach the point where no further detail can be seen. This would have to be the case or the graph would be infinitely detailed. And if an infinitely detailed graph is used to define infinity then we've got a tautology. Therefore we can magnify the graph until we reach the point where no greater detail can be found.
That necessarily means that if x<>0 it is equal to some non-zero number (this statement is redundant but I want to make it very clear). Thus 1/x is always a finite number as long as x is not 0.
Infinity is not equal to 1/0, but 1/x has passed through infinity when x=0.


It couldn't possibly have as 1/x is always a finite number unless x=0. And you said that infinity is not 1/0.

This is simple. As I've previously stated:
My challenge to you is to come up with a definition of 'infinity' that doesn't involve a tautology or subjectivity.
This definition of yours has failed in this regard. Now you can claim:
And I've already demonstrated that any number must be defined in terms of itself, so why demand otherwise of infinity, which is not even a number?


But this is little more than saying all definitions are tautologies so why can't you get away with that when it comes to infinity?
If you've actually demonstrated that any number must be defined in terms of itself then you've proven that numbers cannot be defined objectively. That's a relief. I've always wanted pi to be equal to de jure 4 as the Indiana legistlature attempted to do some years ago. And you also have admitted that you cannot come up with a non-tautological definition of infinity. Therefore you cannot meet my challenge. So why bother giving me a definition of infinity anyway? What's the point?

I do not deny the existence of infinity. I simply claim that it is subjective, not objective, and so it is perfectly acceptable to have a tautological defintion of it. All subjective definitions are like that. Just don't treat infinity as if it were an objective something "out there." It's not "out there." It's within yourself.
This is why infinity is represented by circular things such as a sideways 8 or a cat running around the toilet bowl. It is something that always comes back to itself. And its definition will always be a circular one.
Leave poor Cantor alone, he was no con man.
From Wikipedia

The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.

But Kronecker was right all along. "God made the integers. All else is the work of man."
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Re: Mersenne's Miscelleny

Post by Mersenne »

I'll show you how I know you're wrong instead. Visualize the graph. Imagine that you can increase the magnification of your view until you eventually reach the point where no further detail can be seen. This would have to be the case or the graph would be infinitely detailed.
Which it is.
And if an infinitely detailed graph is used to define infinity then we've got a tautology.
Demonstrate, not define.
Therefore we can magnify the graph until we reach the point where no greater detail can be found.
Non sequitor; your "therefore" does not follow. The tautology is a truth (that's what a tautology is, so to say it's a truth is also a tautology).
That necessarily means that if x<>0 it is equal to some non-zero number (this statement is redundant but I want to make it very clear). Thus 1/x is always a finite number as long as x is not 0.

is not equal to 1/0, but 1/x has passed through infinity when x=0.

It couldn't possibly have as 1/x is always a finite number unless x=0. And you said that infinity is not 1/0.
"Passed through" not "is at". 1/x is not a finite number at every point outside x=0, you're not thinking in terms of limits. Still waiting to be shown!
This is simple. As I've previously stated:


My challenge to you is to come up with a definition of 'infinity' that doesn't involve a tautology or subjectivity.

This definition of yours has failed in this regard.
No it hasn't! "Infinitely divisible" is not the same as "infinite". There is finite space traversed by x, and infinite quantity traversed by 1/x. Suppose we look at x moving from +1 to 0. This is a single finite unit of distance traversed by x. But 1/x will still be infinite at the limit at which it approaches infinity.

EDIT NOTE: I could state that 1/x is infinite when x=0, and this would be true, but it wouldn't be mathematically useful, which is all that is meant by undefined. So I'll continue to argue for the limit, as is conventional.

It's not my definition, just one of those I use.
Now you can claim:

And I've already demonstrated that any number must be defined in terms of itself, so why demand otherwise of infinity, which is not even a number?

But this is little more than saying all definitions are tautologies so why can't you get away with that when it comes to infinity?
Because all definitions are tautologies. Weren't you the one defending definitions as other than tautologies?
If you've actually demonstrated that any number must be defined in terms of itself then you've proven that numbers cannot be defined objectively. That's a relief. I've always wanted pi to be equal to de jure 4 as the Indiana legistlature attempted to do some years ago. And you also have admitted that you cannot come up with a non-tautological definition of infinity. Therefore you cannot meet my challenge. So why bother giving me a definition of infinity anyway? What's the point?

I do not deny the existence of infinity. I simply claim that it is subjective, not objective, and so it is perfectly acceptable to have a tautological defintion of it. All subjective definitions are like that. Just don't treat infinity as if it were an objective something "out there." It's not "out there." It's within yourself.
This is why infinity is represented by circular things such as a sideways 8 or a cat running around the toilet bowl. It is something that always comes back to itself. And its definition will always be a circular one.
Numbers can be correctly defined in terms of themselves (Pi) or incorrectly defined (Indiana Pi). The correct definitions behave consistently with respect to other numbers.
Leave poor Cantor alone, he was no con man.

From Wikipedia

The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.
Kronecker objected to Cantor's proofs on the algebraics and transendentals, then denied their existence. Cantor was correct on both counts- reread your statement, the curriculum follows Cantor not Kronecker. Poincare was not always right, and was wrong here. Cantor was right and incidentally a good man.
But Kronecker was right all along. "God made the integers. All else is the work of man."
I agree. Man has made something, better than God did. *

James, what you're doing is magnificent. You're taking on the whole conventional mathematics establishment on the topic of continuity (not, as you said above, infinity per se). But so far, the establishment's sound.

EDIT: Sorry, that was arrogant. In fact, God got there first.

Suppose time and space are themselves not continuous (quantum physics suggests this). Then they can be modelled with the natural numbers. But they will still exist on a continuum which would have to be modelled by reals (in quantum terms, wormhole space or superspace; the infinite-dimensional and continuous Hilbert space necessary to demonstrate the selfsame graininess of space and time).
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Re: Mersenne's Miscelleny

Post by James Strom »

Non sequitor; your "therefore" does not follow. The tautology is a truth (that's what a tautology is, so to say it's a truth is also a tautology).
It does follow if a tautology isn't acceptable. And it wouldn't be if...
My challenge to you is to come up with a definition of 'infinity' that doesn't involve a tautology or subjectivity.
No justification for a tautology, no matter how good, is permissible without failing the challenge.

Not all definitions involve a tautology if both sides in a debate can agree to some axioms. That's what they're there for. We both can suppose what is otherwise subjective is an objective truth. In this way problems can be resolved and theoroms proven. Otherwisw we're stuck in an endless loop.

Suppose we agreed that if so much as one digit was different in a sequence it cannot be equal to it. For notational purposes we'll assume that ... means "and so on for infinity."

s1 = (0, 0, 0, 0, 0, 0, 0, ...)
s2 = (1, 1, 1, 1, 1, 1, 1, ...)
s3 = (0, 1, 0, 1, 0, 1, 0, ...)
s4 = (1, 0, 1, 0, 1, 0, 1, ...)
s5 = (1, 1, 0, 1, 0, 1, 1, ...)
s6 = (0, 0, 1, 1, 0, 1, 1, ...)
s7 = (1, 0, 0, 0, 1, 0, 0, ...)

s0 = (1, 0, 1, 1, 1, 0, 1, ...)

So there can always be a sequence which isn't on any list and thus the number of sequences is uncountable. O.K.
Let's continue this line of reasoning. If even the change of one digit is enough to make a sequence different then surely the change of two digits would also make them unequal. Can we agree on that? How about all of the digits? If all of them are different then they defintely aren't the same. Right? Right?
Let's look at two numbers in base two, just to keep it simple.
Is 1.0000000... equal to 0.1111111...? It must not be if the above axioms are assumed. And if 0.1111111... is not equal to 1 then it must be slightly less. The only way it could be less is if there were not an infinite number of ones.
Okay, Houston, we’ve had a problem here. Infinity is not infinity.
This is the situation we get into when things aren't strictly defined. In the above example infinity was never defined to begin with. But this isn't surprising. 'Undefined' and 'infinity' are practically synonyms. From an online dictionary:


un·de·fined
[uhn-di-fahynd]

adjective
1.
without fixed limits; indefinite in form, extent, or application: undefined authority; undefined feelings of sadness.

When a motion is tabled "indefinitely" it means that it may be put aside forever and that is infinitely long. Is it any wnder that one divided by zero is called 'undefined'?

Suppose there was a challenge to write a short computer program to print the largest number. They have arbitrarily fast computers. The first guy writes:

10 FOR T=1 TO 65535
20 MEGALITH=2^MEGALITH
30 NEXT T
40 PRINT MEGALITH

But the second guy, thinking he's being clever, writes:

10 INFINITY=1+INFINITY
20 GOTO 10
30 PRINT INFINITY

Who wins? The second number will certainly get, at some point, larger than the first. But it never gets printed. Is the second one even a number? It certainly isn't a definite one.
So maybe that is what is meant when people describe infinity as being 'undefined' or not even a number. Maybe it is a number, just not a definite one. It becomes a process, not a result.
But if it's an indefinite number then it becomes rather useless mathematically. We could prove anything we want using an undefined number.
But infinity does have its uses in the subjective world. Asking the question, "Who am I?" when knowing yourself changes yourself leads to an infinite loop. These sort of things are inevitable when we look at ourselves.
No it hasn't! "Infinitely divisible" is not the same as "infinite".
Yes, it is. If it's infinitely divisible then it has an infinite number of parts. The size of these parts, even if of zero length, is irrelevant.
There is finite space traversed by x, and infinite quantity traversed by 1/x. Suppose we look at x moving from +1 to 0. This is a single finite unit of distance traversed by x. But 1/x will still be infinite at the limit at which it approaches infinity.
If that's not a tautology then I don't know what one is.
Numbers can be correctly defined in terms of themselves (Pi) or incorrectly defined (Indiana Pi). The correct definitions behave consistently with respect to other numbers.
An example of consistency might be pi<>2*pi. However, infinity=2*infinty.
reread your statement, the curriculum follows Cantor not Kronecker.
And therefore Cantor is in the right?
Academia is filled with nonsense theories that serve a more or less philosophical or political purpose but which are otherwise unworkable. Did any biologists dissent from the Lamarckian theory of evolution in the former Soviet Union? Not if they wanted to keep their jobs.
If infinity is treated as something objective then anything can be proven with it. Those who wish to dominate us will often demand that we believe the absurd as a way of showing their power. Think of how Winston Smith was made to agree that 2+2=5 in the novel "1984."
1*infinity =2* infinity. Simplifying we get 1=2. The sky's the limit.
James, what you're doing is magnificent. You're taking on the whole conventional mathematics establishment on the topic of continuity (not, as you said above, infinity per se). But so far, the establishment's sound.
I'm only following in the footsteps of the atomists such as Democritus and Lucretius who just couldn't believe that things could be divided forever. They didn't base their theories on any chemical discoveries.

Suppose time and space are themselves not continuous (quantum physics suggests this). Then they can be modelled with the natural numbers. But they will still exist on a continuum which would have to be modelled by reals (in quantum terms, wormhole space or superspace; the infinite-dimensional and continuous Hilbert space necessary to demonstrate the selfsame graininess of space and time).
If current theories depend upon a continuum to work, and continuums aren't possible in the real world, then the theories are incomplete at the very least.
Remember; if there were such a thing as an infinitely detailed space-time then every point in the universe contains an infinite number of infinitely advanced civilizations within it. All of these would be profoundly affected by anything you did. You never knew how powerful you were.
Or maybe the idea of a continuum is junk science.
Put infinity back where it properly belongs; in the mystical realm.
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