from Mathematical Girl: Gödel’s Incompleteness Theorem
- Syntactic Methods
- Antonym: Semantic Methods
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The difference is whether to use truth values.
- I’m not quite sure.
- In syntactic methods, there is no concept of what is true or false, it’s just about syntax.
- Even if we define something like “axioms,” they are neither true nor false, they are just “axioms.”
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- Understanding it as processing sequences of symbols as sequences of symbols mechanically to do mathematics, like programming (for now).
- This gives me a familiar impression because it’s like programming.
- It’s about “doing mathematics in mathematics.”
- Dealing with formal systems.
- Creating a miniature model of mathematics called “formal system H.”
- Wow~
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Defining the symbol IMPLY: as
- It’s like an alias or wrapper in programming.
- It can be replaced as a string without being interpreted, similar to C++‘s using.
- However, I’m not quite sure why this definition is made.
- Maybe it’s not necessary to think about it when dealing with formal systems, but I want to understand the underlying intention if I want to comprehend it.
- Semantically, if we consider “if a then b,” its truth value matches the truth value of “not a or b.”
- But I still don’t get it.
- Thinking about it in terms of “If apples are delicious, then apple pie is also delicious,”
- Apples are delicious & apple pie is delicious: true
- Apples are delicious & apple pie is not delicious: false
- Apples are not delicious & apple pie is delicious: true
- Apples are not delicious & apple pie is not delicious: true
- Does it mean that anything is okay as long as a is false?
- I can kind of understand it.
- In formal systems, by mechanically executing operations on defined strings, something like “inference” in semantics can be done.
- It is defined as a formal modus ponens
- Antonym: Semantic Methods