My infinity is bigger than your infinity!

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Philosophy versus mathematics:

Mathematicians have taught us that, mathematically speaking, it is useful to talk about different types of infinity. This was a genuine accomplishment.

But whether or not some infinities are "larger" than other infinities is a philosophical question, turning on the concept of infinity itself, and not on any mathematical notions such as one-to-one correspondence.


Comments

  1. Lord3:43 PM

    Countable, philosophic or mathematic?

    ReplyDelete
    Replies
    1. gcallah11:41 PM

      That strikes me as a mathematical concept, but I have not given that much thought.

      Delete
  2. rob8:20 PM

    It's not a matter of philosophy that some infinities are larger than others. There are more real numbers between 0 and 1 than there are integers. I think some dude name Cantor proved this.

    ReplyDelete
    Replies
    1. gcallah11:16 PM

      Amazing, rob! I am ADDRESSING Cantor. What Cantor showed was that the integers cannot be put in one-to-one correspondence with the reals. Yes, I UNDERSTAND that.

      But that does not, necessarily, mean there are "more" reals than integers. Yes, in a FINITE set, it would prove that set A was bigger than set B. But what does it mean in terms of INFINITE sets?

      Sorry, rob, that's a philosophical question.

      Delete
    2. gcallah12:18 AM

      What you have "proved" here, rob, is that you don't actually understand Cantor's mathematical finding!

      Delete
    3. rob7:41 AM

      Understanding what bigger means should not take a philosopher. If you can write an equation to prove something is that philosophy? Infinity is a common part of many mathematical equations.

      Delete
    4. gcallah12:27 PM

      "Understanding what bigger means should not take a philosopher."

      When thing A and thing B are both infinite sets?! Then in fact it DOES take a philosopher!

      Delete
    5. Ken B3:14 PM

      I think the Intuitionist school, Brouwer is the big name, would agree.

      Delete
  3. Keshav Srinivasan10:09 AM

    Gene, "2^Aleph_0 > Aleph_0" is a precise mathematical statement that can be proven in ZFC. Now you can say that the ">" symbol does not capture the philosophical concept of "greater than", but ">" still has a precise mathematical meaning.

    ReplyDelete
    Replies
    1. gcallah12:26 PM

      That's fine, Keshav. I understand the mathematics perfectly well.

      Delete

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