You haven't proven that it can't be proven that 2 + 2 = 5. All you have proven is that if 2 + 2 = 5, then mathematics are inconsistent. Which they could be; you can't just assume math and logic are consistent.
Well, you were responding to someone who was using the correct formal terms. You, after all, don't dispute the second theorem - but given it uses the term "consistent" you can hardly complain when someone makes their argument using the same terminology as the theorem you agree with!
Interested in what you define "consistent" and "valid" as in laymans terms though.
Take two objects. Take two more objects. How many do you have? Four. Could you ever have five? No. Thus, two plus two does equal four, and it does not equal five.
Can you ever prove something to be true, that is not true? No; your proof would simply be incorrect. By definition of "proof" and "truth."
Thus, it can't be proven that two plus two equals five.
I would like to hear your response to this. I don't understand how mathematicians can justify referencing some kind of mathematical formalism and just skipping reality altogether, and then not mentioning that they had done so.
To be clear, I am not contesting Godel's Second Incompleteness Theorem; I'm criticizing the writeup posted here, which uses everyday language to blatantly contradict what is obvious in reality.
For someone who said that someone who pointed out that you can't assume math and logic are consistent "really stepped in it", you have a funny way of proving you don't contest Godel's second incompleteness theorem, which is:
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.
A formalism (layman's definition of "formalism") need not include a statement of its own consistency in order to be valid (including, in order to be consistent).
For example, I know that basic addition and subtraction are valid by observing individual concrete objects in reality, rather than because of some kind of formalism stating that.
I am not very familiar with the kind of "math and philosophy" that went into Godel's theorems, but what I'm saying is that either (a) actual math and logic as such (as opposed to something made up, like a false math where 2 + 2 can equal 5) are ruled out by the giant disclaimer that takes up most of the 2nd theorem (i.e., they are not "formally effectively generated theories..."), or (b) they are, but simply "do not include a statement of their own consistency."
Wow. I've actually done a lot of reading in epistemology and done a lot of thinking about these issues. I would have really appreciated a serious response from you.
For example, I would have liked to hear whether you agree that 2 + 2 = 4 (for example) can be proven from looking at actual concrete objects in reality (e.g., I take 2 pens from the left side of the table, two pens from the right, put them all in the middle, and count them... and then you can, of course, abstract away to get a concept of "unit"... etc).
Try reading up on Hilbert's second problem. Wikipedia does a good job [1]. Then read about Peano axioms [2]. Finally try to understand what is meant by the term "consistency" [3]
