Infinity

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At school, my geometry teacher was talking about how lines can’t be congruent because they extend forever. When asked why lines cant be congruent to one another if all lines extend infinitely, she said that since infinity is indefinite, we can’t really know if one infinity is greater than another. I’m no mathmatician, but isn’t that impossible given the nature of infinity? More importantly, if she is correct, does this in any way effect the theist’s claim that there can’t be an infinite regress of causes? Did she just make a mistake? Is mathmatical infinity somehow different from philosophical infinity?
 
Infinity can come in different cardinalities. You can have different sets of things of the same cardinality and they are “the same size,”, but some cardinalities are bigger.

So, if I remember right, the set of all positive integers is the same cardinality as the set of all (positive and negative) integers. But it’s a different cardinality of the set of all rational numbers, which is different than the set of all reals, which is different than the set of complex. It has to do with whether the sets can be mapped one-to-one to each other. If yes, they are the same cardinality.

Infinity is weird.

But a different cardinality doesn’t negate the issue with an infinite regress of what we’d call an essentially ordered series from actually being. An essentially ordered series must have a first. An accidentally ordered series can be infinite.
 
Milestone said:
At school, my geometry teacher was talking about how lines can’t be congruent because they extend forever. When asked why lines cant be congruent to one another if all lines extend infinitely, she said that since infinity is indefinite, we can’t really know if one infinity is greater than another. I’m no mathmatician, but isn’t that impossible given the nature of infinity? More importantly, if she is correct, does this in any way effect the theist’s claim that there can’t be an infinite regress of causes? Did she just make a mistake? Is mathmatical infinity somehow different from philosophical infinity?
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By “nature of infinity” do you mean that it has no measure?
 
There can be differing mathematical infinities.

Consider the natural numbers, the rational numbers and the real numbers. Each set is “larger” than the one before because it contains the one before, plus an infinity of points not included in the one before. Yet each set is infinite.

(Yes, I know that’s not a proof.)

Philosophical infinity isn’t the same as mathematical infinity because the latter is concerned only with numerical quantification as it grows without bound, whereas philosophical infinity is not limited to the idea of number. (Just from the pigeonholes of my head.)

God Bless and ICXC NIKA
 
Wesrock said:
So, if I remember right, the set of all positive integers is the same cardinality as the set of all (positive and negative) integers. But it’s a different cardinality of the set of all rational numbers, which is different than the set of all reals, which is different than the set of complex.
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Not true.
The set of positive integers, the set of integers and the set of rational numbers all have the same cardinality, and are called countable sets.
The real numbers and the complex numbers have the same cardinality which is different from the cardinality of the integers. There are more real numbers than integers. The number of complex numbers is the same as the number of real numbers because they can be put into a 1-1 onto correspondence.
the cardinality of the integers is less than
the cardinality of the reals which in turn is less than
the cardinality of the set of all functions from R to R.
The continuum hypothesis is that there is no set whose cardinality is strictly between that of the integers and the real numbers.
 
GEddie said:
There can be differing mathematical infinities.

Consider the natural numbers, the rational numbers and the real numbers. Each set is “larger” than the one before because it contains the one before, plus an infinity of points not included in the one before. Yet each set is infinite.

(Yes, I know that’s not a proof.)
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Although the set of natural numbers is strictly contained in the set of rational numbers, they have the same cardinality and are called countable sets. So there are the same number of elements in both sets because they can be put into a 1-1 and onto correspondence.
 
Thanks for the correction. I was going off the top of my head and it’s been awhile.
 
I have question here off topic I guess. But if I have a string 6 inches long, and then make it into a circle, it then has the length of pi-radius-squared. Isnt this a contradiction
 
thinkandmull said:
I have question here off topic I guess. But if I have a string 6 inches long, and then make it into a circle, it then has the length of pi-radius-squared. Isnt this a contradiction
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No. the length of the string is 2r(pi). there is no contradiction if 2r(pi) = 6.
 
pi goes on forever so the length of the circle is different when it is made into a segment
 
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