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Math Help - formal systems help!

  1. #1
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    formal systems help!

    I think I am really confusing myself with this problem. Here it is:

    The formal system A consists of:
    alphabet: 0 /
    axiom: 0
    inference rule: if x is a A-theorem, then so is x/


    It asks for me to state a few theorems of this system. Am I right in thinking that 0/, 00/, 000/, 000//, 00// could all be examples of theorems?

    It continues: Using the theorems of the A-system, we define a new B-system which consists of:
    alphabet: 0,/,[,],>
    axiom: Any A-theorem
    inference rule: if x and y are B-theorems, then so is [x>y]


    It asks again to state a few theorems of this system. Could some examples be [0/>00/], [00//>000/], [0/>0/]

    Any help is really appreciated!
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  2. #2
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    Re: formal systems help!

    Quote Originally Posted by samantha12 View Post
    Am I right in thinking that 0/, 00/, 000/, 000//, 00// could all be examples of theorems?
    No, the inference rule only allows adding /, not 0. So, theorems are 0, 0/, 0//, 0///, etc.

    Quote Originally Posted by samantha12 View Post
    It continues: Using the theorems of the A-system, we define a new B-system which consists of:
    alphabet: 0,/,[,],>
    axiom: Any A-theorem
    inference rule: if x and y are B-theorems, then so is [x>y]


    It asks again to state a few theorems of this system. Could some examples be [0/>00/], [00//>000/], [0/>0/]
    If you keep just one zero on each side of >, then these are theorems. A more interesting example is [[[0/>0//]>[0>0]]>0///].
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  3. #3
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    Re: formal systems help!

    Quote Originally Posted by emakarov View Post
    No, the inference rule only allows adding /, not 0. So, theorems are 0, 0/, 0//, 0///, etc.

    If you keep just one zero on each side of >, then these are theorems. A more interesting example is [[[0/>0//]>[0>0]]>0///].
    Thank you very much! I'm confused as to how you knew that you could only add /s, because the x is not defined? And in the second example again how you knew the definition of x and y?
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  4. #4
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    Re: formal systems help!

    Quote Originally Posted by samantha12 View Post
    I'm confused as to how you knew that you could only add /s, because the x is not defined?
    inference rule: if x is a A-theorem, then so is x/
    The rule says that if one starts from x, one can only add / to it.

    And in the second example again how you knew the definition of x and y?
    One proves the following theorems in sequence. I'll write the corresponding values of x and y from the rule next to the theorem provided it is not an axiom of the second system (i.e., a theorem of the first system).

    Code:
    #1. 0/				Axiom
    #2. 0//				Axiom
    #3. [0/>0//]			x = 0/ (#1), y = 0// (#2)
    #4. 0				Axiom
    #5. [0>0]			x = 0 (#4), y = 0 (#4)
    #6. [[0/>0//]>[0>0]]		x = [0/>0//] (#3), y = [0>0] (#5)
    #7. 0///			Axiom
    #8. [[[0/>0//]>[0>0]]>0///]	x = [[0/>0//]>[0>0]] (#6), y = 0/// (#7)
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  5. #5
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    Re: formal systems help!

    Everything just clicked, thank you so much!
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