# Fun Argument Against Actual Infinites by al-Kindi

#### Torin

##### Well-known member
This is an interesting argument from al-Kindi (the first great Islamic philosopher) which aims to disprove the possibility of an actually infinite body. I do not offer this as a sound proof, just as an interesting thought experiment.

1. Suppose there is an actually infinite body.
2. If we remove one finite part from the infinite body, the remainder after the removal of that finite part is either finite or infinite.
3. The remainder cannot be finite, since then the original body would have been finite and not infinite.
4. So the remainder must be infinite.
5. Now, if we reattach the finite part we removed to the infinite body, the infinite body either increases in size or remains the same size.
6. If it increases in size, then there are two infinite bodies, one smaller and one larger (i.e. from before and after we reattached the finite part).
7. But this means the smaller infinite has limits, which is absurd, since an infinite body has no limits.
8. So the infinite body must remain the same size before and after the finite part is reattached.
9. But this means there is a whole that is not greater than its parts, which is absurd.
10. So the original supposition is false, and an actually infinite body cannot exist.

Thoughts?

• Komodo

#### Algor

##### Well-known member
This is an interesting argument from al-Kindi (the first great Islamic philosopher) which aims to disprove the possibility of an actually infinite body. I do not offer this as a sound proof, just as an interesting thought experiment.

1. Suppose there is an actually infinite body.
2. If we remove one finite part from the infinite body, the remainder after the removal of that finite part is either finite or infinite.
3. The remainder cannot be finite, since then the original body would have been finite and not infinite.
4. So the remainder must be infinite.
5. Now, if we reattach the finite part we removed to the infinite body, the infinite body either increases in size or remains the same size.
6. If it increases in size, then there are two infinite bodies, one smaller and one larger (i.e. from before and after we reattached the finite part).
7. But this means the smaller infinite has limits, which is absurd, since an infinite body has no limits.
8. So the infinite body must remain the same size before and after the finite part is reattached.
9. But this means there is a whole that is not greater than its parts, which is absurd.
10. So the original supposition is false, and an actually infinite body cannot exist.

Thoughts?
Absurdity is the name of the game with the mathematics of infinity. However, the math is internally consistent (the set of all natural numbers, which has an infinite number of members, contains the set of all even, odd and prime numbers, all of which are infinitely large.....)

• Torin

#### Komodo

##### Well-known member
This is an interesting argument from al-Kindi (the first great Islamic philosopher) which aims to disprove the possibility of an actually infinite body. I do not offer this as a sound proof, just as an interesting thought experiment.

1. Suppose there is an actually infinite body.
2. If we remove one finite part from the infinite body, the remainder after the removal of that finite part is either finite or infinite.
3. The remainder cannot be finite, since then the original body would have been finite and not infinite.
4. So the remainder must be infinite.
5. Now, if we reattach the finite part we removed to the infinite body, the infinite body either increases in size or remains the same size.
6. If it increases in size, then there are two infinite bodies, one smaller and one larger (i.e. from before and after we reattached the finite part).
7. But this means the smaller infinite has limits, which is absurd, since an infinite body has no limits.
8. So the infinite body must remain the same size before and after the finite part is reattached.
9. But this means there is a whole that is not greater than its parts, which is absurd.
10. So the original supposition is false, and an actually infinite body cannot exist.

Thoughts?

In #9 al-Kindi is assuming that "the whole must be greater than its parts" is self-evidently true, but this can be doubted when dealing with infinities. Cantor famously argued that the set of all integers, the set of even integers and the set of odd integers were all infinite to the same extent. Or, to put it in more physical terms, suppose we had a ruler which had all the integers, one for every inch. This would of course be an infinitely long body. Now we remove the first three inches. The remainder would still be infinitely long, since the number of integers after 3 is still infinite. If we reattach that 3-inch piece, it still remains infinite. It seems paradoxical to say "therefore it is the same length, before and after," because in all our experience "length" is something finite and measurable, which changes when part is cut away. So if the objection is, "but how can you say it's the same length, before and after the piece was attached?" I think we'd ask what it means to say that two objects are different in length. If it means 'a measuring tape shows...', then that can't be done with the infinite ruler. Cantor said we can compare the sizes of sets by seeing if each element of one could be made to correspond with an element of the other. So "4" for the cut ruler corresponds with "1" for the intact ruler, "5" with "2" and so on. Using that criterion, the two rulers are both infinite and both the same size, though one contains parts that the other does not.

To put it another way, suppose we had two such rulers, one beginning at 0 and one beginning at 3. We lay them side by side, with the 3 of one ruler lined up with the 0 of the other. Is the one starting at 0 longer? In what sense? What if we then erased all the numbers from both rulers: would it still make sense to say that one was longer than the other? No possible measurement could show that difference, could it?

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• Algor

#### Komodo

##### Well-known member
In #9 al-Kindi is assuming that "the whole must be greater than its parts" is self-evidently true, but this can be doubted when dealing with infinities. Cantor famously argued that the set of all integers, the set of even integers and the set of odd integers were all infinite to the same extent. Or, to put it in more physical terms, suppose we had a ruler which had all the integers, one for every inch. This would of course be an infinitely long body. Now we remove the first three inches. The remainder would still be infinitely long, since the number of integers after 3 is still infinite. If we reattach that 3-inch piece, it still remains infinite. It seems paradoxical to say "therefore it is the same length, before and after," because in all our experience "length" is something finite and measurable, which changes when part is cut away. So if the objection is, "but how can you say it's the same length, before and after the piece was attached?" I think we'd ask what it means to say that two objects are different in length. If it means 'a measuring tape shows...', then that can't be done with the infinite ruler. Cantor said we can compare the sizes of sets by seeing if each element of one could be made to correspond with an element of the other. So "4" for the cut ruler corresponds with "1" for the intact ruler, "5" with "2" and so on. Using that criterion, the two rulers are both infinite and both the same size, though one contains parts that the other does not.

To put it another way, suppose we had two such rulers, one beginning at 0 and one beginning at 3. We lay them side by side, with the 3 of one ruler lined up with the 0 of the other. Is the one starting at 0 longer? In what sense? What if we then erased all the numbers from both rulers: would it still make sense to say that one was longer than the other? No possible measurement could show that difference, could it?
(Should have said, "suppose we had a ruler which had all the positive integers.")

#### Electric Skeptic

##### Well-known member
See Hilbert's Grand Hotel paradox for an illustration of some of the absurdities (to us) of infinities. Interesting to contemplate; impossible to comprehend (at least, for me).

• Torin

#### Torin

##### Well-known member
See Hilbert's Grand Hotel paradox for an illustration of some of the absurdities (to us) of infinities. Interesting to contemplate; impossible to comprehend (at least, for me).