Does the below argument make sense?
1. Why say that the following phrase is nonsense?
“If a logical system is consistent, it cannot be complete.”
Because:
The phrase “if a logical system is consistent, it cannot be complete”, is itself a logical system, it is consistent with what it says, and if that is so, something is missing from this phrase, according to what the phrase says. And so this bring us to the second phrase.
2. Why say that the following phrase is nonsense?
“The consistency of axioms cannot be proved within their own system.”
Because:
A system which has axioms for itself, in order for the system to call them axioms for itself, the system has to have a consistent behavior around those axioms and so when it behaves inconsistently with regard to those axioms, the inconsistency between those axioms and the system’s behavior the system can prove to itself.
If what is written above is false, then when a system behaves inconsistently with regard to some axioms it has for itself, that inconsistency it cannot prove to itself, and it keeps behaving inconsistently with regard to those axioms…but…
if the system keeps behaving inconsistently with regard to some axioms and cannot prove to itself that it does so with regard to those axioms, then it doesn’t seem to me it can consistently keep regarding them as axioms for the system, and then something else replaces them, and that something else is what the system calls axioms for itself.
Kind regards,
feel free to remove the post if it doesn't
1. Why say that the following phrase is nonsense?
“If a logical system is consistent, it cannot be complete.”
Because:
The phrase “if a logical system is consistent, it cannot be complete”, is itself a logical system, it is consistent with what it says, and if that is so, something is missing from this phrase, according to what the phrase says. And so this bring us to the second phrase.
2. Why say that the following phrase is nonsense?
“The consistency of axioms cannot be proved within their own system.”
Because:
A system which has axioms for itself, in order for the system to call them axioms for itself, the system has to have a consistent behavior around those axioms and so when it behaves inconsistently with regard to those axioms, the inconsistency between those axioms and the system’s behavior the system can prove to itself.
If what is written above is false, then when a system behaves inconsistently with regard to some axioms it has for itself, that inconsistency it cannot prove to itself, and it keeps behaving inconsistently with regard to those axioms…but…
if the system keeps behaving inconsistently with regard to some axioms and cannot prove to itself that it does so with regard to those axioms, then it doesn’t seem to me it can consistently keep regarding them as axioms for the system, and then something else replaces them, and that something else is what the system calls axioms for itself.
Kind regards,
feel free to remove the post if it doesn't
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