Suppose that the truth is what is (the optimistic version) or would be (the counterfactual version) known (by beings like us) in the ideal limit. In both cases, it seems there are things I know that aren't true, which is absurd. For instance I know that on the table beside my laptop there is right now a water bottle arranged thus-and-so relative to a used wet wipe.
Or was. For I just threw out the wet wipe, and moved the bottle. In a week or a year, most likely I'll forget how these were arranged. It's already starting to fade a little. I don't expect to tell anybody. So, here is something I knew: The water bottle and the wipe were arranged thus and so at 8:17 am on September 2, 2009. Is this something that would be known in the ideal limit? I doubt it. While the powers of science will grow in the progress to the ideal limit, the facts about the water bottle and the wipe will recede into the past, and their traces will be covered up. For a while, the fact could be pulled out from my brain by careful investigation of the memory structures. But presumably eventually the brain will rot (unless the Second Coming comes first). It may be true that the rotted matter will be slightly differently arranged for this memory. But the difference will be harder and harder to discover as time goes on.
Suppose this argument succeeds. And suppose that I accept the ideal limit theory of truth. Then I should say: "I know p but p is not true." And a theory of truth that implies that is absurd.
There are two ways out of this predicament for the ideal limiter. The first is to deny that the ideal limit for all true propositions is reached at the same time, at the culmination of science. The ideal limit for some propositions, such as that the items on the table were arranged thus-and-so, is reached much earlier, say, now. One way to try to do this is to say that the ideal limit has been reached for p at t provided that p is believed by someone at t, and in the course of progress towards an ideal science, an undefeated defeater for p will never be found. An obvious problem, however, is that I might form a false belief about some really trivial matter, then forget the belief, and the course of ideal science would never be able to recover the situation from the rotting matter of my brain to provide a defeater. A further problem is with conjunctions. For let q be some proposition that is known only when science is completed. Then, p and q are both true, and hence their conjunction is true. But their conjunction is never known.
The second, which Jon Kvanvig offered me, is to say that the ideal limit involves time travel. We even have an argument for the possibility of time travel: Truth is ideal-limit knowledge; it is true that the bottle and wipe were arranged thus-and-so; therefore that they were arranged thus-and-so is known in the ideal limit; the only way this could be known in the ideal limit is by time travel; hence, time travel is possible. Now there seems to be something very fishy about a theory of truth that, when conjoined with trivial observations, implies the scientific claim that time travel is possible. Moreover, it is surely true that nobody is time traveling to my home on 8:37 am on September 2, 2009. But if in the ideal limit there were such time travel, as the hypothesis suggests, then we have a truth that isn't true: namely, that nobody is time traveling to my home on 8:37 am on September 2, 2009.