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Gödel, Escher, Bach: An Eternal Golden Braid,
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Roger Merritt
An axiom is something that is just assumed to be true. It is not based on logic or reasoning. It is said to be self-evident. The truth of it is supposed to be so obvious that it need not be proven. Indeed, it cannot be proven.
A theorem is an assertion of fact. It usually is not obvious. Its truth can only be proven by arguing according to a set of rules called "logic." Its truth depends on the truth of the premises, so if you can start with axioms and follow the rules and reach the assertion, then the theorem is considered "proven," and "true," and can be used as the basis for further argument to prove other theorems.
A theorem is an assertion of fact. It usually is not obvious. Its truth can only be proven by arguing according to a set of rules called "logic." Its truth depends on the truth of the premises, so if you can start with axioms and follow the rules and reach the assertion, then the theorem is considered "proven," and "true," and can be used as the basis for further argument to prove other theorems.
Phượng Minh
Axiom means "absolute truth" - We accept it without condition.
Theorem is a statement with logic and argument with a proof based on a system of consequences of axioms.
Theorem is a statement with logic and argument with a proof based on a system of consequences of axioms.
William Keely
Typically the only way to prove an axiom, is to prove that it can not be proven. Therefore, they are "assumed" facts. Theorems are built off of axioms and their properties.
Nicolay
I see axioms and theorems explained with their traditional meanings.
But in GEB, there is a TNT (typographical number theory), and the meaning is much simpler.
The most important thing: there is a set of rules to derive a string of characters based on another string of characters.
In TNT, an axiom is a string present on the set of valid strings from the start, it was not derived from another string.
TNT theorems are strings derived from the other strings (axioms and previous theorems) using the set of rules.
But in GEB, there is a TNT (typographical number theory), and the meaning is much simpler.
The most important thing: there is a set of rules to derive a string of characters based on another string of characters.
In TNT, an axiom is a string present on the set of valid strings from the start, it was not derived from another string.
TNT theorems are strings derived from the other strings (axioms and previous theorems) using the set of rules.
uosɯɐS
An axiom is considered to be a "true statement" from the start. Theorems are added to the pile of "true statements" as one moves about the system, using the system's rules on the existing pile of "true statements," in order to generate more "true statements."
In typical logic, certain rules and axioms are assumed to hold. But for Hofstadter, Gödel, et al, "the system" need not be standard logic. The idea is to learn about the nature of systems themselves by playing with different systems.
In typical logic, certain rules and axioms are assumed to hold. But for Hofstadter, Gödel, et al, "the system" need not be standard logic. The idea is to learn about the nature of systems themselves by playing with different systems.
Jim Lindsay
Axiom -- something that is almost universally accepted to be true, is almost obvious, but is not provable. E.g.: On a plane, parallel lines never touch each other.
Conjecture -- something that is believed to be true, but is not yet proven. See: https://en.wikipedia.org/wiki/List_of...
Theorem -- something that is proven. E.g.: Pythagorian Theorem
Conjecture -- something that is believed to be true, but is not yet proven. See: https://en.wikipedia.org/wiki/List_of...
Theorem -- something that is proven. E.g.: Pythagorian Theorem
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