Conversation started Oct 9, 2017 at 14:13.
user131753
Oct 9, 2017 14:13
Sure but before that please go through @Conifold's answer to the following question,
user131753
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Q: What sources discuss Russell's response to Gödel's incompleteness theorems?

user 170039In his book My Philosophical Development Russell writes, In my introduction to the Tractatus, I suggested that, although in any given language there are things which that language cannot express, it is yet always possible to construct a language of higher order in which these things can b...

philosophy-of-mathematics reference-request bertrand-russell goedel
user131753
Especially the following, "If we add the negation of the Godel sentence to Peano arithmetic Godel's proof presumably produces a contradiction, yet this can not be the case since then the sentence would be provable. So one of the axioms that "must be false" would be one from "school-boy's arithmetic", and clearly "this is not what is intended". Therefore, one needs a "language of higher order in which these things can be said", according to Russell."
user131753
I interpreted the above part of the answer in the following way, "Suppose that $\sf{PA}$ is consistent. Now consider ${\sf{PA}}+\neg \text{Con}({\sf{PA}})$. If it is inconsistent, then in every model of ${\sf{PA}}$ it is true that $\text{Con}({\sf{PA}})$. So by Completeness Theorem it follows that ${\sf{PA}}\vdash \text{Con}({\sf{PA}})$. But this contradicts Gödel's Second Incompleteness Theorem.
user131753
Hence ${\sf{PA}}+\neg \text{Con}({\sf{PA}})$ must be consistent. However I don't really understand how from this it follows that (as @Conifold writes in his answer) "[s]o one of the axioms that "must be false" would be one from "school-boy's arithmetic"". Can you explain that to me @user21820?
user131753
Oct 9, 2017 14:36
I have to go now @user21820. Take your time. (Honestly, I can intuitively understand how it can be said informally that one of the axioms that "must be false" would be one from "school-boy's arithmetic" but I want a more precise and formal answer. I hope my question is not trivial)
user21820
user21820
Oct 9, 2017 16:00
@user170039 The answer is: Godel's proof obviously does not produce a contradiction in the formal system PA+¬Con(PA), as you should know, and so everything after the first claim in what was apparently quoted from Russell is invalid.
Anyway these people are no longer around for us to probe them, so let's not bother about what they said anymore, except perhaps to find inspiration.
I say this because there seems to be no end to new generations claiming that people misunderstand somebody long dead, which to me seems like idolization in many cases, though we clearly can't ever prove it one way or another. My personal viewpoint is that if logicians in that time make criticism of a non-logician misunderstanding something in logic, it is 90% likely to be valid criticism. Part of the reason is that they had opportunity to present their criticism in logically clear form.
In nearly all cases they didn't, which in my personal view shows that they didn't understand what they were talking about, whether or not due to not wanting to understand all the technical details.
 
Conversation ended Oct 9, 2017 at 16:06.