Infinity always fascinated me in my math studies. The concept itself is hard enough to wrap one's mind around, but that there was more than one kind (even excluding the car) and some were "bigger" than others can hurt your brain.
The easiest one to understand is the set of counting numbers (known also as whole numbers, integers, cardinal numbers) which are infinite. Think of a straight-edge marked in inches. Extending its length forever we still could count each inch: 0,1,2,3,4,... That set is countably infinite. A mind-blower here: extending our forever straight edge in the other direction, into negative number territory we can count 0,1,-1,2,-2,3,-3,4,... so we have another countably infinite set, which is the same level of infinity as the first one!
Now let's put marks between the inches dividing each into tenths, then divide each of the tenths into ten smaller parts, then each of those by ten... The set of all those marks on our endless straight edge is infinite but uncountable, because each division can be further divided, adding ever more hash marks. In fact - and here's the second mind-blower - the set of possible divisions between 1 and 2 is infinite and uncountable, as are the possible divisions between 1 and 1.1, and they are the same level of infinity!
There are more infinities than those two, but if I haven't already lost you, going any further would guarantee it, so let's talk about something else.
How about the Universal Declaration of Human Rights?
This is a fascinating document, produced by the newly formed United Nations in 1948 in an attempt to establish some ethic and moral base for governance. It was approved forty-eight to zero with eight abstentions. Two covenants further clarifying human rights were signed in December 1966:
Economic, Cultural, and Social Rights
and
Civil and Political Rights.
It's interesting to see the signatories who have yet to ratify the covenants they signed: China and Cuba on Political Rights; South Africa and the USA on Social Rights.
Taken all together the three documents seem to me to net out to the famous Four Freedoms spoken of by FDR and beautifully illustrated by Norman Rockwell. You can find them on the net easily if you're interested.
I guess the "something else" had a bit of a political edge to it. Sorry, but I get bothered when I see so many human rights violations being accepted even promoted by people who, I would hope, know better: my countrymen.
The easiest one to understand is the set of counting numbers (known also as whole numbers, integers, cardinal numbers) which are infinite. Think of a straight-edge marked in inches. Extending its length forever we still could count each inch: 0,1,2,3,4,... That set is countably infinite. A mind-blower here: extending our forever straight edge in the other direction, into negative number territory we can count 0,1,-1,2,-2,3,-3,4,... so we have another countably infinite set, which is the same level of infinity as the first one!
Now let's put marks between the inches dividing each into tenths, then divide each of the tenths into ten smaller parts, then each of those by ten... The set of all those marks on our endless straight edge is infinite but uncountable, because each division can be further divided, adding ever more hash marks. In fact - and here's the second mind-blower - the set of possible divisions between 1 and 2 is infinite and uncountable, as are the possible divisions between 1 and 1.1, and they are the same level of infinity!
There are more infinities than those two, but if I haven't already lost you, going any further would guarantee it, so let's talk about something else.
How about the Universal Declaration of Human Rights?
This is a fascinating document, produced by the newly formed United Nations in 1948 in an attempt to establish some ethic and moral base for governance. It was approved forty-eight to zero with eight abstentions. Two covenants further clarifying human rights were signed in December 1966:
Economic, Cultural, and Social Rights
and
Civil and Political Rights.
It's interesting to see the signatories who have yet to ratify the covenants they signed: China and Cuba on Political Rights; South Africa and the USA on Social Rights.
Taken all together the three documents seem to me to net out to the famous Four Freedoms spoken of by FDR and beautifully illustrated by Norman Rockwell. You can find them on the net easily if you're interested.
I guess the "something else" had a bit of a political edge to it. Sorry, but I get bothered when I see so many human rights violations being accepted even promoted by people who, I would hope, know better: my countrymen.
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Comments are always from "anonymous". Often I can identify the author by the content of the comment, but that much cogitation makes my 80 year-old brain tired. Please help out an old man and identify yourself within the text of the comment. Thanks for the comments whether or not you ID yourself. Tom