- Oct 2, 2011
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William Lane Craig wrote:
Mathematically, a series is the sum of the terms of a sequence. The natural number series diverges to infinity. This means that as you add more terms, the sum grows larger and larger without approaching any finite limit. I have no idea what he means by "the natural number series is complete".
Further, when I first read the above question, I experienced anterior cingulate cortex dissonance. His question made no sense to me.
If you use math terminology, it is better to stick to the technical definitions.
Now, the question becomes:
But I don't think that's what Craig had in mind.
Let i1 be a natural number. Is it possible to append i1 to the infinite sequence of all natural numbers, such that i1 has not appeared before?
No, by definition.
Perhaps, that wasn't what Craig had in mind either.
Finally, is it possible to add (i.e., adjoin) a new natural number to the set of all natural numbers, thereby changing its cardinality?
No. I think Craig also had trouble distinguishing different orders of infinities.
Craig conflated the math concepts of series with sequence—and probably with the concept of set too. By misusing terminology, Craig risked confusing readers and weakening the rigor of his reasoning.
Bold emphases added.
Mathematically, a series is the sum of the terms of a sequence. The natural number series diverges to infinity. This means that as you add more terms, the sum grows larger and larger without approaching any finite limit. I have no idea what he means by "the natural number series is complete".
Further, when I first read the above question, I experienced anterior cingulate cortex dissonance. His question made no sense to me.
If you use math terminology, it is better to stick to the technical definitions.
Now, the question becomes:
Yes, if the sequence is finite. A sequence is not a set.
But I don't think that's what Craig had in mind.
Let i1 be a natural number. Is it possible to append i1 to the infinite sequence of all natural numbers, such that i1 has not appeared before?
No, by definition.
Perhaps, that wasn't what Craig had in mind either.
Finally, is it possible to add (i.e., adjoin) a new natural number to the set of all natural numbers, thereby changing its cardinality?
No. I think Craig also had trouble distinguishing different orders of infinities.
Craig conflated the math concepts of series with sequence—and probably with the concept of set too. By misusing terminology, Craig risked confusing readers and weakening the rigor of his reasoning.
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