INFINITY: EXPLORING THIS ENDLESS ENIGMA

A rather large quantity. And a concept we may have encountered, and possibly struggled with, in an occasional math course.

But why bother talking about it? ทดลองเล่นสล็อต pragmatic play Infinity hardly seems relevant to the practical matters of our normal day, or even our abnormal days.

Well, possibly, but infinity does pose a high intellectual intrigue. So a few minutes with infinity should provide a strong mental challenge and a diversion from the tribulations of our normal day. At least enough to warrant a few minutes consideration.

And dismissing infinity as irrelevant misses at least one relevant aspect of the concept.

God.

Believer or not, searcher for faith or not, detester of the concept or not, God, whether as an object of faith, or an ultimate question, or an irrational delusion, God looms as unavoidable. God either serves as guidance for our life, or poses questions bedeviling our minds, or lingers as an outmoded concept born of ancient history in pre-scientific times.

And a major tenet in most theologies, and in philosophy in general, points fundamentally to an infinite God – infinite in existence, infinite in knowledge, infinite in power, infinite in perfection.

So as a passing, but intriguing, diversion, and as an attribute of a spiritual figure deeply imbedded in our culture and our psyche, infinity does provide a subject worth a few minutes of our time.

So let’s begin.

How Big is Infinity?

Strange question, right. Infinity stands as the biggest quantity possible.

But let’s drill down a bit. We should apply some rigor to examining infinity’s size.

Consider integers, the numbers one, two, three and up, and also minus one, minus two, minus three and down. We can divide integers into odd and even. Common knowledge.

But let’s consider a not-so-obvious question, a question you might have encountered. Which is larger, all integers, or just even integers? The quick answer would say the group of all integers exceeds the group of even integers. We can see two integers for every even integer.

If we have studied this question previously, however, we know that answer is wrong.

Neither infinity is larger; the infinity of all integers equals the infinity of just even integers. We can demonstrate this by a matching. Specifically, two groups rank equal in size if we can match each member of one group with a member of the other group, one-to-one, with no members left over unmatched in either group.

Let’s attempt a matching here. For simplicity, we will take just positive integers and positive even integers. To start the match, take one from the set of all positive integers and match that with two from the set of all positive even integers, take two from the set of all positive integers and match that with four from the set of even positive integers, and so on.

At first reaction, we might intuit that this matching would exhaust the even integers first, with members of the set of all integers remaining, unmatched. But that reflexive thought stems from our overwhelming experience of finite, bounded sets. In a one-to-one matching of the rice kernels in a two pound bag with those of a one pound bag, both finite sets, we well expect the one pound bag to run out of rice kernels before the two pound bag.

But infinity operates differently. An infinite set never runs out. Thus even though a one-to-one matching of all integers verses even integers runs pragmatic play up the even integers side quicker, the even integers never run out. Infinity presents us features counter-intuitive to our daily experience filled with finite sets.

And so with fractions. The infinite set of all fractions does not exceed the infinite set of all integers. This really throws a counter-intuitive curve, since we can not readily devise a one-to-one matching. Would not the fractions between zero and one loom so numerous that no matching can be created? But that would be wrong.

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