Are we too free with infinity?

[mistermack]

"Argument is an intellectual process. Contradiction is just the automatic gainsaying of any statement the other person makes."

No it isn't.

[colubridae]

"Argument is an intellectual process. Contradiction is just the automatic gainsaying of any statement the other person makes."

No it isn't.

Mandelson? Is that you?

[Xamonas Chegwé]

Infinity is not a number. In that sense, you are right when you say that 'an infinite number' is a meaningless phrase.

Infinity is a shorthand way of saying that something has no boundaries, that it keeps on going forever.

And infinity does exist in the real world. eg. This argument is infinite!

[mistermack]

Infinity is not a number. In that sense, you are right when you say that 'an infinite number' is a meaningless phrase.

That right. If there actually was such a thing as an infinite number, and there was an infinite number of numbers between one and two, then each one would have a predecessor, and the difference would be zero. That's the infinity of nonsense I meant.
The nonsense is introduced by the contradiction between 'infinite' and 'number'.
You can't have an infinite number of numbers, and the numbers can't have a predecessor, because they don't exist. But if they did exist, they would. It goes round and round ad infinitum.
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[JimC]

So for the case of real numbers, we always have a procedure for producing a third real number between any other real number. For prime numbers, we always have a means of producing a larger prime number. For directions, we always have a means of identifying a direction that is not identical to any previously given direction.

That illustrates what I'm saying really. What we have is a procedure, or a means. What we don't have is an infinite number of anything.
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It depends what you mean by "have". In the case of examining the real numbers that could be written that are larger than 1, but smaller than 2, some elementary logic will show that such a list could be continued indefinitely; that is to say, it is not possible to exhaustively list all the real numbers in that set.

Infinity is simply a useful shorthand for that concept. Any confusion only arises when we use the term "infinity" in a context where it does not belong.

[mistermack]

That's what it boils down to really, we have a procedure that has no known cut off point. The infinite number of numbers don't exist, but it's possible to keep generating them.
That's why I suggest that phrases like 'there is' or 'there are' or 'we have' are incompatible with the word 'infinity'.
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[JimC]

That's what it boils down to really, we have a procedure that has no known cut off point. The infinite number of numbers don't exist, but it's possible to keep generating them.
That's why I suggest that phrases like 'there is' or 'there are' or 'we have' are incompatible with the word 'infinity'.
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There has been a long debate in the mathematical community as to the existence of "actual" infinities as opposed to "potential" infinities.

I actually take the view that they do exist, at least in an abstracted world of possible entities, a somewhat platonic world-view I know...

If I take an idealised straight line segment within this platonic world, I must conceive that it is composed of an infinite number of points, as represented by co-ordinates along its length.

The argument as to whether any possible set of entities in the physical universe can be truly represented by infinity is a more fraught one...

[mistermack]

I agree, but it's an infinity of nothing, which is easy enough to agree to.
An infinity of something is much harder to find.
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[ChildInAZoo]

That right. If there actually was such a thing as an infinite number, and there was an infinite number of numbers between one and two, then each one would have a predecessor, and the difference would be zero. That's the infinity of nonsense I meant.

Even if there are no infinities, what you have written there is still nonsense.

The cardinality of the set of rational numbers (that is, fractions) between 0 and 1 is the same as the cardinality of the set of all whole numbers. That is, we can construct a procedure to match up all the whole numbers with all the rational numbers between 0 and 1, each of these rational numbers distinct from all the others by a finite amount.

And there is a very well-understood sense in which there are more real numbers between 0 and 1 than there are rational numbers.

[mistermack]

It's nonsense, I said it's nonsense, it's the logical consequence of nonsense.
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[ChildInAZoo]

It's nonsense, I said it's nonsense, it's the logical consequence of nonsense.
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Fallacy ad mistermack: arguing that because you think something is nonsense, even if you haven't studied it, it is nonsense.

[Xamonas Chegwé]

Infinity is not a number. In that sense, you are right when you say that 'an infinite number' is a meaningless phrase.

That right. If there actually was such a thing as an infinite number, and there was an infinite number of numbers between one and two, then each one would have a predecessor, and the difference would be zero. That's the infinity of nonsense I meant.
The nonsense is introduced by the contradiction between 'infinite' and 'number'.
You can't have an infinite number of numbers, and the numbers can't have a predecessor, because they don't exist. But if they did exist, they would. It goes round and round ad infinitum.
.

Here is where you are wrong.

It is not possible to count the numbers between 1 and 2. They are uncountable. They are unbounded. They are infinite. This is not a number - this is a shorthand way of saying that they cannot be counted - that they go on forever. That no matter how many numbers you can name between 1 and 2, I can name, not just one more, but as many more as I like.

1.5 is between 1 and 2. So is 1.55. So are 1.555, 1.5555, 1.55555, 1.555555, etc. However many 5s you add to the end of that decimal, I can add one more and it will be a different number. How infinite do you want? And I chose decimals consisting of strings of 5s at random. I could have chosen fractions.

1 and a half, 1 and a third, 1 and a quarter, 1 and a fifth, 1 and a sixth... How far do I need to go before the numbers are 'the same'? Answer - never. You can keep shrinking the fractional part for as long as you like but you will always get a unique number every time.

The set of numbers between 1 and 2 is a continuum. It has no gaps. It is a line, filled from start to finish with numbers. Zoom in to any magnification and the numbers are as tightly packed as at any other. No number has a predecessor but each has its place in the line and they are all different.

[mistermack]

Ok Xamonas, but you've replied to only part of what I wrote.
You've ignored the phrases ''If there actually was such a thing'', and ''You can't have an infinite number of numbers, and the numbers can't have a predecessor, because they don't exist.''.

[colubridae]

Infinity is not a number. In that sense, you are right when you say that 'an infinite number' is a meaningless phrase.

That right. If there actually was such a thing as an infinite number, and there was an infinite number of numbers between one and two, then each one would have a predecessor, and the difference would be zero. That's the infinity of nonsense I meant.
The nonsense is introduced by the contradiction between 'infinite' and 'number'.
You can't have an infinite number of numbers, and the numbers can't have a predecessor, because they don't exist. But if they did exist, they would. It goes round and round ad infinitum.
.

Here is where you are wrong.

It is not possible to count the numbers between 1 and 2. They are uncountable. They are unbounded. They are infinite. This is not a number - this is a shorthand way of saying that they cannot be counted - that they go on forever. That no matter how many numbers you can name between 1 and 2, I can name, not just one more, but as many more as I like.

1.5 is between 1 and 2. So is 1.55. So are 1.555, 1.5555, 1.55555, 1.555555, etc. However many 5s you add to the end of that decimal, I can add one more and it will be a different number. How infinite do you want? And I chose decimals consisting of strings of 5s at random. I could have chosen fractions.

1 and a half, 1 and a third, 1 and a quarter, 1 and a fifth, 1 and a sixth... How far do I need to go before the numbers are 'the same'? Answer - never. You can keep shrinking the fractional part for as long as you like but you will always get a unique number every time.

The set of numbers between 1 and 2 is a continuum. It has no gaps. It is a line, filled from start to finish with numbers. Zoom in to any magnification and the numbers are as tightly packed as at any other. No number has a predecessor but each has its place in the line and they are all different.

Don't forget an infinite number of transcendentals.

[ChildInAZoo]

Ok Xamonas, but you've replied to only part of what I wrote.
You've ignored the phrases ''If there actually was such a thing'', and ''You can't have an infinite number of numbers, and the numbers can't have a predecessor, because they don't exist.''.

No, what he did was demonstrate that what you had written doesn't make sense.