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Thread: Natural numbers

  1. #1 Natural numbers 
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    Scientists say that the universe may not be infinite at all. It could curve itself back from where it started. Infinity gets a knock here.
    If so why assume natural numbers to be infinite? Can't they too reach a saturation point?


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  3. #2  
    Forum Masters Degree thyristor's Avatar
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    The natural numbers follow Peanos axioms, Universe doesn't.


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  4. #3 Re: Natural numbers 
    Forum Professor sunshinewarrior's Avatar
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    Quote Originally Posted by Joseph
    Scientists say that the universe may not be infinite at all. It could curve itself back from where it started. Infinity gets a knock here.
    If so why assume natural numbers to be infinite? Can't they too reach a saturation point?
    The boundedness of the universe is a matter of physical fact, about which we may make abstract speculations though.

    The infiniteness of the natural numbers is a mathematical fact, made true in absolute (read tautological) terms by the axioms and definitions of mathematics - not, therefore, subject to physical evidence or confirmation.

    Ergo, the analogy between the two idea of infinity and limitation, does not actually hold.
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  5. #4  
    Forum Radioactive Isotope MagiMaster's Avatar
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    It's actually pretty simple to see why the natural numbers must be infinite. Assume otherwise. Then there must be a largest number. But then I could just add one, which would be a new largest number, so my original number wasn't the largest. But this applies to any number including my new largest number. That's a contradiction: there's a number larger than the largest number. So the only conclusion is that there isn't a largest number, i.e. numbers are infinite.
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  6. #5  
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    Quote Originally Posted by MagiMaster
    It's actually pretty simple to see why the natural numbers must be infinite. Assume otherwise. Then there must be a largest number.
    Define "larger", and "largest"
    But then I could just add one,
    Define "add". You are making intuitive assumptions you are not yet (in your "proof") entitled to make.
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  7. #6  
    Forum Radioactive Isotope MagiMaster's Avatar
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    True, but that wasn't meant to be a rigorous proof. Of course, it's pretty easy to see where you should go from there to make it rigorous (or at least, it seems like it).
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  8. #7  
    Forum Masters Degree thyristor's Avatar
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    Quote Originally Posted by MagiMaster
    True, but that wasn't meant to be a rigorous proof. Of course, it's pretty easy to see where you should go from there to make it rigorous (or at least, it seems like it).
    Look up Peano's axioms if you don't know of them :-D
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  9. #8  
    Forum Masters Degree thyristor's Avatar
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    Quote Originally Posted by MagiMaster
    True, but that wasn't meant to be a rigorous proof. Of course, it's pretty easy to see where you should go from there to make it rigorous (or at least, it seems like it).
    Look up Peano's axioms if you don't know of them :-D
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  10. #9  
    Forum Radioactive Isotope MagiMaster's Avatar
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    Yeah. I know of them. I just meant that it seems like it should be an easy proof, but you never know until you try.
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  11. #10  
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    Quote Originally Posted by Guitarist
    Quote Originally Posted by MagiMaster
    It's actually pretty simple to see why the natural numbers must be infinite. Assume otherwise. Then there must be a largest number.
    Define "larger", and "largest"
    But then I could just add one,
    Define "add". You are making intuitive assumptions you are not yet (in your "proof") entitled to make.
    Your criticism would be appropriate in the thread on the Peano Axioms. But here the subject started with the natural numbers and one might reasonable assume that the notion of ordering and of addition of natural numbers are known. MagiMasters proof of the fact that there is no largest natural number within that context is perfectly all right.
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