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Top 10 blatant contradictions in math

[This blog post was originally a standalone webpage I wrote sometime in 2016. I edited it a tiny bit for the blog. It's satire, intended to poke fun at lists of "contradictions" in the bible. The apparent contradictions in these theorems can all be resolved.]

Sorry, matheists, but every thinking person admits that abstract objects don't exist. Math is just a fairy tale invented by professors to control their grad students. Pathetically, the fairy tale isn't even internally consistent:

1. A theorem by Shamir says that \(\mathbf{IP} = \mathbf{PSPACE}\), but a theorem by Chang et al. says that there exists an oracle \(A\) such that \(\mathbf{IP}^A \neq \mathbf{PSPACE}^A\).

Every middle school student knows that if you do the same thing to both sides of an equation, it stays an equation! So much for matheists' doctrine of "theorem inerrancy".

2. Bell's theorem says that through quantum entanglement, objects can instantaneously affect each other across arbitrary distances. But the no-communication theorem says the opposite: acting on a quantum system has no effect on any entangled systems.

What's more, the no-communication theorem is just plagiarized from physicists! One of the laws of special relativity says you can't send a signal faster than the speed of light. Sound familiar?

3. Supposedly, 0.999... = 1.

If matheists learned how to use logic, they would immediately see how silly this is.

4. Taylor's theorem says that every infinitely differentiable function can be expanded as a Taylor series, but the Taylor series of \(\exp(-1/x^2)\) around zero is identically zero.

Matheists only buy this stuff because of scare tactics. They're told that if they don't believe the theorems, they will not receive eternal employment.

5. A theorem by Johnson says that it takes \(\exp(\Omega(d))\) books to build a stack that leans over distance \(d\). But a theorem by Paterson and Zwick says \(O(d^3)\) books suffice.

Don't those dumb matheists realize that polynomials grow slower than exponentials?

6. The law of large numbers says that if you keep sampling a probability distribution, the sample average converges to the expectation. But a theorem by Poisson says that if you sample from the standard Cauchy distribution, the sample average still follows the standard Cauchy distribution.

More like the rule of thumb of large numbers, am I right?

7. Cayley's theorem says that every finite group is a subgroup of \(S_n\) for some \(n\). That means there are only countably many finite groups. But there are uncountably many distinct regular polygons (since there are uncountably many side lengths by a theorem of Cantor), and each has a finite group of symmetries.

Connect the dots, sheeple. It's not that hard.

8. Gödel's completeness theorem says that the standard axioms of logic are enough for any proof; no axioms are missing. But Gödel's incompleteness theorem says that there does not exist a complete list of axioms; there are always missing axioms.

The contradiction is right there in the names of the theorems! Maybe matheists would see it if they weren't so brainwashed by the sermons they're always listening to in classrooms.

9. A standard theorem of Euclidean geometry says that if two solids are congruent, then they have the same volume. But the Banach-Tarski paradox says that you can cut a unit ball into five pieces and rearrange them into two disjoint unit balls.

Let me guess: this theorem was "taken out of context"?

10. Kolmogorov's zero-one law implies that either 100% of numbers have terminating decimal expansions or else 0% of numbers have terminating decimal expansions. But some numbers (like 1/2) do and others (like 1/3) don't.

Remember, just because some old guys published some proofs, that doesn't make a theorem absolute truth.