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One Plus One Equals Two (2006)

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114 points lemper | 2 comments | 22 Oct 24 08:21 UTC | HN request time: 0.437s | source
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earthboundkid ◴[22 Oct 24 18:32 UTC] No.41917250[source]▶
Wait, am I crazy for thinking relations are not sets? Two sets can be coextensive without the relation have the same intension, no? Like the set of all Kings of Mars and the set of Queens of Jupiter are coextensive, but the relations are different because they have different truth conditions. Or am I misunderstanding?
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JadeNB ◴[22 Oct 24 19:23 UTC] No.41917740[source]▶
> Wait, am I crazy for thinking relations are not sets? Two sets can be coextensive without the relation have the same intension, no? Like the set of all Kings of Mars and the set of Queens of Jupiter are coextensive, but the relations are different because they have different truth conditions. Or am I misunderstanding?

No-one can stop you from using terms as you please and investigating their consequences, but, at least in modern mathematical parlance, a binary relation is the set of ordered pairs that are "related" by it. (Your relation would seem to be just a bare set, or perhaps a unary relation, not a binary relation which I think is what is often meant without default modifier.)

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1. cubefox ◴[23 Oct 24 13:55 UTC] No.41925093[source]▶
He is talking about the difference between intension and extension. The properties "creature with a heart" and "creature with kidneys" are different, even though they may have the same extension (if the set of creatures with a heart and the set of creatures with a kidney happen to be the same). This also applies to relations of arbitrary arity. In mathematics everything is usually treated as extensional, because all the mathematical objects, like numbers, exist "necessarily". This is not the case for other objects, where things could be the same (like the set of creatures with heart and the set of creatures with kidneys) but they aren't necessarily the same. It's possible that there is a creature with heart but without kidneys. Though even in mathematics, properties that define the same objects are often not trivially equivalent: they are necessarily equivalent, but it may take a complex proof to show that they are.
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2. earthboundkid ◴[26 Oct 24 19:04 UTC] No.41956943[source]▶
She.