1) You can sell your house for $0.
2) If someone would buy your house for some detetrmined price, they will also buy it for that much money plus 1 cent.
3) By the Mathematical Induction, it's now proven that you can sell your house for as high of a price as you wish.

you haven't proven either of your premises.

2) does not work. A person has a limited amount of money and a limited amount of money they can borrow. If all the money I can use to buy something is say, $2000, then I cannot pay $2000.01

SPOILERS

Both 1 and 2 are self-evident and pretty much true. True enough for them to work. (Not for argument as a whole to work though.)

If all the money you can use to buy something IS REALLY 2000, in all but the most exceptional circumstances, you can just go on a street and beg for a cent.

Also, you don't want to buy ANYTHING for a price that equals the maximum amount of money you can access.


What really fails is the point number 3. The use of mathematical induction there is incorrect. Why ?

This is what logical loop M.I. creates:

0.0) There exists a person that can buy your house for $0. (See 1)
1.1) If we ask that person for $0.01 more, he'll still buy it. (See 2)
1.2) Therefore, there exists a person that can buy your house for $0.01.
2.1) If we ask that person for $0.01 more, he'll still buy it. (See 2)
2.2) Therefore, there exists a person that can buy your house for $0.02.
3.1) If we ask that person for $0.01 more, he'll still buy it. (See 2)
3.2) Therefore, there exists a person that can buy your house for $0.03.

The lines which index ends on 1 are not supported by practically correct statements 1 or 2.
They are, in fact, false.
Just because a person can be convinced to pay a slightly larger price once, it doesn't mean that there exists a person initially willing to pay that price. And a single person can agree to a slightly increased price only for a finite, limited amount of increases.

In a proper context for M.I., of course, if after a change of A to B it continues to satisfy something, B, as in "just B", is also going to satisfy it. For example, 4 is always the same 4, regardless of whether if was created by 2+2, 2x2, 2^2, or 1+1+1+1.

I've known people who have had to pay to get someone else to take a house off of their hands. The premise that anyone will buy your house for $0 is not true at all, because there are various financial and legal liabilities involved.

Considering the solution posted, it might be more accurate to have named the thread "Quick Logic Problem".
A maths problem would be concerned about the soundness of the argument, that is, are the premises true, and does the conclusion have to be true based on those premises? While a logic problem is merely concerned with if the premises are true, does the conclusion have to be true as well?

Thread renamed.
You can replace $0 by $1 (and maybe dance around arithmetic a bit) and still have the same problem.
The argument implies an incorrect "hidden premise" that mathematical induction applies to states of a person as well as it applies to states of different people.
Or something close to that.

see, this is the problem: you're using hard mathematical induction when your premises are just "true enough". you can't do that: induction requires axiomatic truth. the thing must always be true, with no caveats or exceptions. Premise 1 can be replaced with "you can sell your house far enough under market value that someone will buy it" in order to work around potential legal issues that may arise from a $0 sale, but Premise 2 just doesn't hold in the face of scrutiny. there are too many circumstances where that just isn't true, so you can't use induction.

note that the end result of this is roughly comparable to your conclusion: convincing someone to go up is different from them initially being willing to pay that price. the reason this breaks your system, though, is not because mathematical induction fails from your premises (it doesn't) but because your premises are flawed. the equivalence of the two states is baked into Premise 2. so you have the right conclusion, but you've misidentified the point of failure.

i think you'll find that for every penny you add, especially past a certain point, people become very very slightly less willing to buy it