Contradictions in infinity

Maybe this won’t be as deep as it sounds, but here goes.

-Space is infinite, and in infinicy, everything that is possible has to happen sometime, right?
-There are many things that are equally possible, and yet mutually exclusive (the various Gods for example).
-Thus, if the universe is infinite, then these mutually exclusive, yet equally possible things must all exist at one point or another.
-The universe is not infinite, because you can’t have 2 mutually exclusive things existing at the same time.

Example: Both Allah and Yahweh can’t both exist at the same time, because they are both the only “true” gods, and they are both omniscient and omnipotent and time-transcending. And yet since they are equally possible, they would both have to exist at the same time, given that the universe is infinite, and they are both everywhere at once. Umm… I think I have confused myself.

Do you kinda see what I am getting at here though?

Space might not be infinite, but not because logic dictates it.

Atheists will no doubt be along any moment to point out that you can have as many imaginary deities as you like.

Could you perhaps think of another pair of mutually exclusive infinites?


An infinitly long string of numbers consisting entierly of repeating ones will never have a two in it, even though two is a possible number.

Infinite space doesn’t imply that everything that can exist does.

Well Check I am forced to say a few things here…First of all The universe is infinite because if it were finite there would have to be a boundry at the end…and I do not believe that is possible. Second, the human mind with all its complexity has to make-up and believe in higher powers, to give us spacial cognition and a place on this planet, I would venture to say when religion was forming people really liked the idea of someone else outside of themselves taking responsibility for the actions of others… i.e. God/the Devil made me do it… But that does not prove that the tooth fairy exists just because my little girl wants her too…If there is a God and ‘HE/SHE/IT’ has always existed and always will exist, then I guess you just described the going definition of the universe itself now didn’t you?

The universe could well be finite but boundless (like the surface of the earth (to tread out a well-worn analogy) - although you have to consider the earth’s surface to be a warped 2D plane for this to make any sense.

No, AndrewL is right. Think of tossing a fair coin 10 times. Does “tails” ever have to occur? No, of course not. What about 1000 or 1,000,000 times? Tails still does not have to occur, nothing about the coin requires that. In fact, there is some possible universe accessible to our own where no fair coin has ever landed on tails.

Well, only an atheist (who denies all gods) would claim that all gods are equally possible (with probability of existence 0). Any believer knows that a particular God has probability of existence of 1 and all others of 0. :wink:

I think you’re confusing “possibility of belief” with “possibility of existence”.


Obviously, you can’t claim this since none of your previous claims are true. Your reasoning is still provocative. I wonder if you can actually prove the proposition that “2 mutually exclusive things [cannot exist] at the same time”, other than by assuming its truth?

kg m²/s²

My last paragraph doesn’t exactly say what I wanted it to say. The question is, can you prove that there necessarily is a pair of objects with the property “cannot exist at the same time”? Are there a pair of objects such that one exists at time t in all possible worlds, and the other does not exist at time t in any possible world?

If you flip a coin n times the odds of getting at least one heads increases as n increases. This doesn’t mean that when the probability of getting at least one heads in n trials reaches one that we must suddenly get a heads, though.

In an infinite universe it would seem that everything could happen has probability of 1 for occuring at least once somewhere. Of course, this still doesn’t mean in must happen, and in an infinite universe the probability of anything happening in a certain portion of space should be zero, unless the universe is infinitely old, too, in which case the probability for anything occurring at any portion in space at least once is 1.

Mixing logic with theism = not a good idea.

This is most likely incorrect, which invalidates your entire argument.

I’m afraid I don’t. Are you trying to prove that the universe is finite? I thought it was the general consensus of cosmologists that the universe probably IS finite, so your supposed paradox wouldn’t exist. And if it isn’t finite, how would infinite space equal “everything that we can imagine must happen”?

Sorry, that was supposed to be one coherent thought rather than 2 posts. Please ignore the first one - I would delete it if I were able to.

Here’a contradiction of infinities;

Consider two sets of infinite numbers that we’ll call A and B. A is the set (1,2,3,4,5,…); B is the set (10,20,30,40,50…). Now compare the sets:


As you can see, it’s clear that every member in A can be exactly matched to a member in B. So the two sets must have an equal number of members.

Now let’s make a new set called C. C is the set (5,10,15,20,25,…). Compare A and C.


You can easily see that A contains all the members of C and for every member in C, A contains four other numbers as well. A clearly has more members than C.

Now compare C and B.


By the same argument as the previous comparision, you can easily see that C contains all the members of B and for every member in B, C contains another number as well. C clearly has more members than B.

So now what have we proven?
A has more members than C.
C has more members than B.
A and B have the same amount of members.

<< An infinitly long string of numbers consisting entierly of repeating ones will never have a two in it, even though two is a possible number. >>

A more telling example: the decimal 0.101001000100001…
(There’s a 1 and then one 0; a 1 and then two 0’s; a 1 and then three 0’s; a 1 and then four 0’s; etc. forever).

This is an infinite, non-repeating decimal… but it not only doesn’t have a two in it, it doesn’t have 3, 4, 5, 6, 7, 8, or 9 in it.

So, the statement that in an “infinity”, every possibility must occur is just not so.

Little Nemo: your example is not a contradiction, but a misunderstanding of how set-theory and infinite collections can be “counted.” All three of your sets are the same “size” even though one is contained in the other. The problem comes from thinking about “infinity” as though it were a finite number.

You have an infinite set (say the set {1,2,3,4,…}) and you add one more element (like 0). Your mind, used to thinking about finite numbers, thinks the new set {0,1,2,3,4,…} is larger than the old set by one additional element, but this is not so. The sets are the same size; for every element n of the one set there is a corresponding unique element n-1 of the second set; and for every element of the second set m, there is a corresponding unique element m+1 of the first set. So the two sets are equal in size, since they can be put into one-to-one correspondance, each with the other.

That you can’t count :slight_smile:

Two sets are equinumerous if there exists an isomorphism between their elements. The existence of other, non-isomorphic mappings is irrelevant.

By your logic, I can take the sets {dog, cat, bird, mouse, iguana, turtle, tarantula} and {red, yellow, blue, green}, and a mapping where every pet is mapped to red; and conclude that four colors are more than seven pets. After all, there are colors left over that aren’t assigned to a pet.

CK Dexter, permit me to suggest that you have missed the numerical translation of the OP.

Consider a random number generator, or if you prefer, the infinite digits of PI.

Given that 1) In a random infinite space any possible item will be represented. (This is true, and one of the better random number generator tests)


  1. That an infinite amount of some number, let’s say 1, is a possible item.

  2. Any random string of numbers …34621… must have an set of infinite members, i.e. an infinity of 111111’s.

Paradoxical? Yes, but not fully – they’re called lesser and greater infinities, which is what little nemo illustrates. Not convinced? Consider an pillar, 8 feet around, stretching to vertical and horizontal infinity. Can I not split this pillar into many pillars, say 1 foot around? Yes, and each will be infinite and smaller than the original infinity. The original pillar represents an infinite random set and the smaller infinities are easily contained within it. And yet there are things that it may not contain, for instance, something 9 feet around.

I think you’re confusing the notion of an infinite universe with multiple (or parallel) universes where our ‘laws’ of nature might not apply. Logic (in our universe) dictates that two mutually exclusive things can not be present, regardless of how ‘infinite’ the universe is (yes, I know that the notion of ‘how’ infinite is a contradiction! :smiley: )
However, in an (infinite) parallel universe model, anything that COULD be possible IS going to turn up somewhere. Thus in universe X-6 we have Allah, and in univ. 23.Y7 we have Wowbagger wandering around abusing everybody.

And of course, as has been mentioned, we are not even sure whether this universe is infinite or not!

Is this true? As I’ve said in a different thread, I’m more or less innumerate, but I was led to believe that that’s not how probability works. I was told it’s a common misconception of how probability works, but not the way it actually does.

I was taught that the universe has no memory, so each individual coin toss is a separate event. Every toss has a 50% chance of falling heads up. No matter if you toss the coin 10 times or 10,000,000,000 times.

Correct me if I’m wrong, please. I try to make my math knowledge correct, even if it’s not that extensive.


There are actually different sizes of infinite sets (countable vs. uncountable). George Cantor, the German 19th century mathematician, oddly proved this using his famous diagonal process. I explained the idea in another thread (Anything Goes). I’ll just quote it here:

Bijection, not isomorphism. You need some sort of algebraic structure for an isomorphism, which isn’t neccesary for establishing the cardinality of a set.

Also, other mappings aren’t neccesarily irrelevant. As long as they’re injective (no two elements are mapped to the same point) or surjective (for every element in the set being mapped to some, not necessarily unique, element in the initial set maps to it) you can deduce inequalities for the cardinalities. (Incidentally, for those who don’t know and care a bijection is a map which is surjective and injective)

The flaw in his argument is the fact that you can establish a mapping like that doesn’t show that A has more members than C, it shows that A has more than or equal to members than C. (And similarily for the other parts).

erislover: Errr. You never have a probability of 1 or greater that a head has occured. In the coin tossing experiment the probability of a head occuring at least once is 1 - 1/2^n, which is always less than 1. You can never have probabilities greater than 1. Further, the fact that the probability of an event occuring is 1 does in fact mean the event must occur.