Actually, the Martingale system does pay off if you have unlimited resources. However casinos artificially create limited resources by setting table limits.

Table limits exist to segregate gamblers based on betting patterns, except where set by law. A pit boss will almost always allow you to place a bet over the limit if you clear it with him first and, if not, will direct you to the nearest table with higher limits.

They allow this precisely because the Martingale system does NOT pay off,

From wikipedia:

"It is therefore a good example of a Taleb distribution – the gambler usually wins a small net reward, thus appearing to have a sound strategy. However, the gambler's expected value does indeed remain zero because the small probability that he will suffer a catastrophic loss exactly balances with his expected gain."

The quote is out of context: In case of infinite reserve (and infinite time to place your bets ;)), there are no catastrophic losses which would stop you from betting again.

The problem with always doubling your wager is that the exponential growth will eventually reach any finite limit in case of a run of 'bad luck'. Without limit, that's not an issue and you'll only lose when you never win any bet, which 'almost surely' won't happen.

Correct, I skimmed the Wikipedia article too quickly. It is a neutral outcome.

The definition of losing in gambling is running out of resources. If you stipulate infinite resources, you're stipulating that you can't lose. In which case, clearly, you have a much better chance of winning.

There are two mistakes in your reasoning.

First, according to your definition of losing, gambling away 99% of your life savings would not count as a loss. A better definition would be: having fewer resources at the end of an act of gambling than you had before the act.

Second, you have assumed that infinite comes in only one size. To see the error in that, consider these infinite sets: {1,2,3,...} and {2,4,6,...}. The first set contains all counting numbers and the second set contains only even counting numbers. Both sets are infinite, but the second set is only half the size.

With these corrections in mind, it should become clear that you can start with infinite resources, lose half of your resources and therefore lose at gambling, but still walk away with infinite resources.

As cbr pointed out, the two sets are equinumerous (y_n = 2 * x_n puts them into 1:1 correspondence). As long as you gamble a finite number of times, and your losses are finite, you will always have exactly as much money as you started with, if you started with infinite money. Since there is no such thing as infinite money, this model is only a practical approximation when your losses are greatly dwarfed by your starting capital.

I should have been more careful in choosing my words. I agree that {1,2,3,...} and {2,4,6,...} are both countably infinite and have the cardinality aleph-0. When I said they are not the same size, I did not mean to invoke the idea of set cardinality. Rather, I was thinking more in terms of set difference: if set A contains all the elements of B and set A contains other elements also, then set A is bigger than set B.

With that said, the reasoning that you two are employing seems mysterious. In my way of thinking, the cardinality of a set of dollar bills and the quantity of dollars are not the same thing. If you start with $20 and lose $20, then you lost $20, and likewise, if you start with an infinite quantity of dollars and lose $20, then you lost $20. Whether the set of dollars before and after gambling have the same cardinality is quite beside the point: $20 never equals $0, so you were $20 richer before you gambled and $20 less rich after you gambled.

I disagree with your claim that "[the set of integers] is bigger than [the set of even numbers]." They have the same cardinality, and I'm not sure how else you can reasonably define "bigger" or "size" for infinite sets. If you prefer to treat it like real money, you can buy just as much with {1, 2, 3,...} dollars, and {21, 22, 23, ...} dollars (after you lost $20). Or, from another direction, you could go to a currency exchange and do a real life bijection, trading 20 -> 1, 21 -> 2, 23 -> 3, and so on. Either way, it makes sense to say you have the same amount of money. "Makes sense," since we're talking about an abstraction that doesn't exist.

If mathematically, the sets are the same size, and practically, in terms of buying power, the bankrolls are the same size, I'm not sure how we can reach an abstraction in either perspective where one is "less" than the other.

(The question of walking away is sort of ill-formed for the original Martingale discussion -- since the point there was that you'd stay at the table until you won a spin.)

Interestingly this brings up another simple flaw in the "infinite resources" caveat. When you do finally win a spin, you'll still have aleph-null dollars -- you won't have actually won anything. So the only condition under which the martingale works -- when you can't lose -- is one under which you can't win either. Unless your goal is not to increase your purchasing/gaming power but to break the casino, as in an "Ocean's N" film.

  > When I said they are not the same size, I did not mean
  to invoke the idea of set cardinality.

Set size and cardinality are one in the same. You would have to change the axioms of set theory in order to define size any other way.

  > if set A contains all the elements of B and set A
  contains other elements also, then set A is bigger than
  set B

Not if you take "bigger" to mean "of greater size." This statement is only true if "is bigger than" means "includes" (= "is a superset of"). Set inclusion and set size are distinct concepts. We can say that if set A contains all the elements of set B, then set A is at least as numerous as set B. But we cannot say that if set A additionally contains elements that set B does not, then set A is more numerous than set B.

  > In my way of thinking, the cardinality of a set of
  dollar bills and the quantity of dollars are not the
  same thing.

They are certainly distinct concepts, but for a countably infinite set of bills with bounded face values, the two quantities are equal (even if they are computed differently).

  > if you start with an infinite quantity of dollars
  and lose $20, then you lost $20

Let us distinguish money, the sum of the nominal value, from currency, the specific expression of that value. If you lose $20, then your currency has changed (let's say you lost a specific $20 bill), but your money has not (you can keep losing $20s without any meaningful consequence).

  > Whether the set of dollars before and after gambling
  have the same cardinality is quite beside the point: $20
  never equals $0, so you were $20 richer before you
  gambled and $20 less rich after you gambled.

How do you define "rich"? Do you define "rich" to be "in possession of a large quantity of money" or do you define "rich" to be "in possession of a specific set of dollar bills"? They are not the same. I don't care if my $20 bill has serial number X or serial number Y, it's still worth $20 to me.

Likewise, although you can measure the value of the difference between the set of money you had before and the set of money you have after, as long as that difference is finite, it has no bearing on your infinite total value. In fact, depending on how you do it, you can lose an infinite amount of money and still have an infinite amount left (the difference between {1, 2, 3, ...} and {2, 4, 6, ...} is {1, 3, 5, ...} and all three of these sets are equinumerous).

Infinity is not an intuitive concept.

While there are larger and smaller infinite sets the two sets you give are the same size. They are both countably infinite. A way to see this is that for every element in the set of even positive integers there is a corresponding positive integer and vice versa.

All math is abstraction, and you need to make sure it applies to the current case. In this case it doesn't: infinite resources don't exist.