Is every statement true?

wait... brain... hurt... YES!

Here's the flaw in your argument...

It comes at the point where you consider an arbitrary false statement and try to show that it is a member of the set of true statements. (Nothing wrong with doing that in logic, if you can pull it off, but here you just can't...)



Okay you're considering the case of a statement that is false.

In "proof by contradiction", an assumption that a false statement is true leads to a contradiction, from which in turn follows any conclusion you like, including your own desired conclusion in this thread that our false statement is a member of the set of true statements.

But we are not assuming that our false statement is true; we are considering the case where it is false. A contradiction does not follow from an assumption we haven't made about our false statement being true. We may not conclude anything we like from a contradiction that we haven't derived. Specifically, we cannot conclude that the false statement under consideration is a member of the set of true statements.

Sorry, friendo, but your argument doesn't demonstrate that there is any kind of paradox associated either with proof by contradiction or with axiomatic set theory.

It does, however, demonstrate some clever sleight of hand.

This doesn't contradict the fact that a false statement implies a contradiction, as indirect proofs show. Any statement we consider can be thought of as having a "...is true" suffix added to it. For example, "1 + 1 = 2" can be thought of as saying "1 + 1 = 2 is true," and "1 + 2 = 5" can be thought of as saying "1 + 2 = 5 is true." When we consider a statement we thus are always considering its truth. Consider the statement "The color red is green." This is like saying "'The color red is green' is true." This statement is false, because it implies a contradiction. Not only does it imply a contradiction, but it can be thought of as being a contradiction in and of itself. After all, red is not green.

When you say you are saying essentially the same thing. When we assume a false statement, we are assuming the statement as if the "... is true" suffix was there. While the statement may be false, it itself says it is true. Not only does this "lead" to a contradiction, it's a contradiction itself. As you can see here we do not even have to "assume" the statement true at all. A false statement, in and of itself, is a contradiction. Thus it's false, but it's also true. So, every statement is true, while only some of these statements are false.

the existence of the quality of every statement is true. if a statement is false, then the falsity of that statement exists, and is therefore true.

yes, and no.