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The solution to induction
Posted: 29 June 2011 01:09 PM   [ Ignore ]   [ # 61 ]
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Write4U - 29 June 2011 12:59 PM
DarronS - 29 June 2011 06:00 AM

W4U, infinity is not a number, it is a concept, therefore you cannot perform mathematical operations on infinity.

From my computer’s online dictionary, in mathematics infinity is:

a number greater than any assignable quantity or countable number

Is that not what I originally posited?

Infinity is always 1 (at least) larger than the largest (countable) number.

I stipulate that the number 1 may be debatable mathematically, but the idea was “larger than”.

The definition is not the same as what you posted. What you posted makes use of the notion of “the largest (countable) number” which is senseless. There is no such number, and indeed it would create a contradiction if there were.

And if you are thinking that the ‘number’ infinity is 1 (or some countable number) larger than some other countable number, you’re not understanding infinity. Even a countable infinity (the smallest order of infinity) is infinitely larger than any other countable number. So if (per impossible) you had “the largest countable number”, you would have to add infinity to it to get to infinity.

For more on infinity check out the wiki on infinity. I haven’t read the whole thing but it looks pretty good.

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Posted: 29 June 2011 01:18 PM   [ Ignore ]   [ # 62 ]
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Let me put it another way. It sounds to me like you’re saying infinity is like this: “Take any number, the largest you can think of. Well, infinity is larger than that number.”

That’s true.

But where it goes off the rails is where you think that this means all you need to do to get to infinity is add some number to the largest number you can think of, and there you are. But what you really have to do is to add infinity to it, i.e. the whole number line. It’s the whole line that’s infinitely long, not any (finite) part of it.

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Posted: 29 June 2011 01:25 PM   [ Ignore ]   [ # 63 ]
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Has anybody seen the BBC documentary called “Dangerous Knowledge” on Cantor and others ho studied infinity and uncertainty?  I was called “Dangerous Knowledge.”  It was really good.  Apparently many mathematicians who studied infinity went insane.  The documentary is on YouTube HERE.

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Posted: 29 June 2011 01:36 PM   [ Ignore ]   [ # 64 ]
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dougsmith - 29 June 2011 01:18 PM

Let me put it another way. It sounds to me like you’re saying infinity is like this: “Take any number, the largest you can think of. Well, infinity is larger than that number.”

That’s true.

But where it goes off the rails is where you think that this means all you need to do to get to infinity is add some number to the largest number you can think of, and there you are. But what you really have to do is to add infinity to it, i.e. the whole number line. It’s the whole line that’s infinitely long, not any (finite) part of it.

IMO, the keyword in my posit was “always”. No matter how large the number, infinity is always (1) larger (ad infinitum).

I think I do understand the concept, I just did not use the formal language. And I will stipulate to that assertion.

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Posted: 29 June 2011 01:52 PM   [ Ignore ]   [ # 65 ]
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Write4U - 29 June 2011 01:36 PM

IMO, the keyword in my posit was “always”. No matter how large the number, infinity is always (1) larger (ad infinitum).

I think I do understand the concept, I just did not use the formal language. And I will stipulate to that assertion.

OK, well, it sounds as though infinity is sort of hovering a few digits ahead of your large number all the time, which isn’t the right thought. Instead it’s the “ad infinitum” that’s the crucial step there. If you keep adding on infinitely, the whole thing will be (a countable) infinity. But then it makes no difference if you add 1 or 4 or 1,234,565 or a googol each time.

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Posted: 29 June 2011 02:12 PM   [ Ignore ]   [ # 66 ]
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dougsmith - 29 June 2011 01:52 PM
Write4U - 29 June 2011 01:36 PM

IMO, the keyword in my posit was “always”. No matter how large the number, infinity is always (1) larger (ad infinitum).

I think I do understand the concept, I just did not use the formal language. And I will stipulate to that assertion.

OK, well, it sounds as though infinity is sort of hovering a few digits ahead of your large number all the time, which isn’t the right thought. Instead it’s the “ad infinitum” that’s the crucial step there. If you keep adding on infinitely, the whole thing will be (a countable) infinity. But then it makes no difference if you add 1 or 4 or 1,234,565 or a googol each time.

I understand, I used the number 1 as a theoretical quantity.

But I believe what I posited is in agreement with Hilbert. His hotel has an infinite number of rooms, filled with an infinite number of guests (i.e. it is full), yet there is always room for 1 more guest.

[ Edited: 29 June 2011 02:48 PM by Write4U ]
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Posted: 29 June 2011 05:33 PM   [ Ignore ]   [ # 67 ]
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brightfut - 28 June 2011 03:31 PM
dougsmith - 28 June 2011 03:59 AM

  (1) All even numbers are divisible by two

I know that all even numbers are divisible by two, but how do we know that all even numbers are divisible by two?  I mean, what is this mathematical reasoning called?

I think that is an axiom also. i.e. by definition, an “even number” is a number that is divisible by two.

There is no reasoning involved.

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Posted: 29 June 2011 07:32 PM   [ Ignore ]   [ # 68 ]
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domokato - 29 June 2011 05:33 PM

  I think that is an axiom also. i.e. by definition, an “even number” is a number that is divisible by two.

There is no reasoning involved.

Good point   wink  Doug may have meant to refer to the phenomenon that we know that any number no matter how big that ends in 0,2,4,6, or 8 will be divisible by 2.  This has been proven.

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Posted: 29 June 2011 07:50 PM   [ Ignore ]   [ # 69 ]
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brightfut - 29 June 2011 07:32 PM
domokato - 29 June 2011 05:33 PM

  I think that is an axiom also. i.e. by definition, an “even number” is a number that is divisible by two.

There is no reasoning involved.

Good point   wink  Doug may have meant to refer to the phenomenon that we know that any number no matter how big that ends in 0,2,4,6, or 8 will be divisible by 2.  This has been proven.

Right. grin

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Posted: 30 June 2011 08:19 AM   [ Ignore ]   [ # 70 ]
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Point of fact; definitions are also the product of human reason and are subject the same nuances and limitations as other products of the same.

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Posted: 30 June 2011 11:28 AM   [ Ignore ]   [ # 71 ]
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I’m not sure all definitions require “reason”. What reasoning is required to define “even numbers” as “those which are divisible by two”?

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Posted: 30 June 2011 11:54 AM   [ Ignore ]   [ # 72 ]
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domokato - 30 June 2011 11:28 AM

I’m not sure all definitions require “reason”. What reasoning is required to define “even numbers” as “those which are divisible by two”?

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Posted: 05 August 2011 07:41 PM   [ Ignore ]   [ # 73 ]
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Stormy Fairweather - 25 June 2011 07:45 AM

The only things it is reasonable to accept on faith, to believe in, is our existance, awarenes of it and capacity to understand. ...

I would remove the word “only” form the above statement. There are many more things we accept on faith, for example, existence of other people, existence of many galaxies, existence of atoms, etc.

[ Edited: 05 August 2011 08:05 PM by Ludwik Kowalski ]
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Posted: 05 August 2011 08:58 PM   [ Ignore ]   [ # 74 ]
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Ludwik Kowalski - 05 August 2011 07:41 PM
Stormy Fairweather - 25 June 2011 07:45 AM

The only things it is reasonable to accept on faith, to believe in, is our existance, awarenes of it and capacity to understand. ...

I would remove the word “only” form the above statement. There are many more things we accept on faith, for example, existence of other people, existence of many galaxies, existence of atoms, etc.

IMO “faith” is a very vaporous word. It relies only on trust, hope, and belief in that which is not provable.

This is why the term usually applies to religious beliefs. It is the hope and belief that something does exist in spite of lacking any evidence of its existence. All things spiritual are accepted on faith.
We do have evidence that other people exist, that there are many galaxies, that atoms have been observed by scientists. To me that does not strictly qualify as faith.
Perhaps we exercise faith in the existence of virtual particles, however even here we have indirect evidence of their existence.
Even faith in another person’s ability and/or integrity is usually based on knowledge (evidence) of that person. Thus is a projection of how this person will act in the future based on past experience with that person.

Interestingly, no such past experience applies to religious faith. It is not founded on any empirical evidence. It is inductive reasoning based on an unknowable premise.

[ Edited: 06 August 2011 06:12 PM by Write4U ]
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