"no-self defeating object" Here, Terence Tao presents a collection of similar mathematical arguments that he calls "no self-defeating object" (examples are Euclids proof of the infinitude of the primes and Cantor's theorem). In the second post, he remarks that one can reformulate these "no-self defeating object" arguments to get a "every object can be defeated"-version.
The simplest example "no self-defeating object" goes as follows:

Proposition 1 (No largest natural number). There does not exist a natural number N that is larger than all the other natural numbers.
Proof: Suppose for contradiction that there was such a largest natural number N. Then N+1 is also a natural number which is strictly larger than N, contradicting the hypothesis that N is the largest natural number.

The corresponding "every object can be defeated"-version is:

Proposition 1′. Given any natural number N, one can find another natural number N' which is larger than N.
Proof. Take N' := N+1.

My question: What does Tao mean by an object being "defeated"? Does he mean that "defeating" means that a certain object doesn't have the property? But then, what does it mean that "an object defeats itself"?
 A: It sounds to me like "no self-defeating object" arguments are just a form of proof by contradiction. One assumes that a certain object exists, then derives a contradiction from the assumption of its existence, thereby showing that the object does not actually exist. 
Here, the object is "defeated" because it is shown not to exist. And it is self-defeated because the defeat (the contradictory conclusion) follows from postulation of the very object itself.
One can often recast a proof by contradiction into a direct proof---that is, one in which the conclusion follows directly by reasoning or rules of inference from some premises (without having to assume the negation of the conclusion you want to show, as in proof by contradiction). 
In your example, one thinks: maybe N is my candidate greatest natural number. But N fails in this, by considering N+1. N's presumptive status as greatest natural number is defeated, not by deriving a contradiction from N's presumptive status itself, but simply by its successor; and every candidate N is susceptible to this argument. Hence, "every object is defeated."
"Defeat" is just a silly metaphor and I wouldn't too get hung up on it, but I take it that this is the intuition.
