Apparently, not even what you will find written in any book about elementary set theory (as actually it is taught to older children) will convince you. If you have kids, check out their little math book. That is usually on page one or two. The basics, really.
I think you're lying... go ahead and snap a picture of a little math book that discusses an empty set and shows that it is included as subset of any set. Please also include the page number.
that is what you will find in any book of set theory, no matter how elementary. No matter how childish, or done with little cute pictures so that also kids can get it. This is the basic of mathematics and logic. This is something known for centuries and part of the education of older children.
Nope. I think you're lying. Please bring a cute little picture of an empty set included as subset. Here's what I found.
Notice what's said at the bottom. This is important. "In practice this can be used to make proving a statement easier". That is the root of the problem.
if you think otherwise, then I suggest you show me the evidence, for instance in the form of a publication that says empty sets cannot be subsets, or recreate the entire edifice of mathematics and logic with your ideas. However, I am not holding my breath.
Well that's the problem right there. It's a mismatch in values. It cannot be proven that an empty set ISN'T a subset. But that doesn't mean that an empty set IS a subset of everything. You seem to be valuing a lack of proof and then elevating that to a position of truth. In other words, just because something cannot be proven to be false, that doesn't make it true. An immoral person cannot see this.
No. This is not logic. Logic is a method for developing true conclusions. What you are describing does not reject conclusions with zero truth. That's what makes it immoral. It's lying about what it is. And that's why it permits a statement like "All the Jews I know are Atheists" even if the speaker doesn't know any Jews. The system is corrupt. And if that is what is being taught to children, it is teaching them to lie. It is teaching them wrong.
Listen to yourself, you contradict yourself in 1 second without any qualms or reservations.
it is not the case that subset A being a subset of B entails that they must have elements in common. This is just what you made up. Unless you can show me where you got that nonsense.
See below:
the definition is quite simple: A is a subset of B if each element of A is also an element of B.
There you have it. My definition is coming from you. And you said, set A does not entail that set B have elements in common. And then you said A is a subset of B if each element of A is also in B. That is an obvious contradiction.
It does not say anywhere that A must have elements.
Yes, it does, you just said it.
A is a subset of B if each element of A is also an element of B.
There it is. The definition requires that A has an element. If it lacks this an element, then it does not saticfy the condition.
That is the property P that must be fulfilled by all elements of A, in order for A to be subset of B.
OK, so what happens if A has no elements? Let's see what you say next.
However, if A is the empty set, then property P is trivially fulfilled by all elements of A, because, as we have seen, for any property, the elements of the empty set fulfill that property, trivially, since it has no elements.
As we have seen, how? Where? Listen to yourself, it's empty, but it fulfills. That's nonsense. Do you even know how this is derived? You don't get it do you? This is dogma, an axiom, and it is based on redefining truth in a way which is unexceptable to any moral person.
Ergo, the empty set is a subset of every set, including itself
Ergo, this is a false proof. It's not trivially fulfilled. It's not fulfilled at all. It's called "vacuous" in the wikipedia article you quoted multiple times yeterday. Of course, eventhough that was included in the middle of what you quoted, you intentionally cropped it out. Another sign of the morally bankrupt. A null set does not obtain all. That is absolute rubbish. It is not something that has been "seen to be true. It's assumed to be true because it cannot be proven false. That is an immoral version of true.
Here's how this actually works. And if you go through and understand the derivations, any and all of them, you'll see I'm right.
The defintion you brought for the subset is true. But the question is, how does one make that determination.
The straight forward approach is correct. One looks at set A and compares the elements in it to set B. If all the elements in A correspond to the elements in B without any remaining, without any extra elements in A, then A is a subset of B. That's a straight forward test based on objective truth. The elements exist, I can count them and compare them. And this conforms to the defintion you provided. No one is making up a new defintion. It works everytime even with an empty set. The reason this is the correct method is because it is complete. It looks for both matching elements and non-matching elements. The result is a conclusion which is demonstrably true. It has truth, it has value.
There's also a counter intuitive approach which is incorrect. It's essentially a short-cut. It's incomplete. As I noted above, using the contrapostive is often easier, but it comes with a price. For this, the contrapostive works in all cases except with an empty set. Instead of the straight forward test, one could do the exact opposite of the previous approach and try to prove that A is NOT a subset of B. The so-called logic of this is, if it cannot be proven not to be a subset, then it must be a subset. To do this, one starts with the elements of B comparing each one to the elements in A to see if A has any elements that B doesn't have. And this works for all cases excluding the empty set. For the empty set, he comparrison cannot be done, and therefore it cannot be shown there there is an element in A which is not in B and this renders a false positive to the test. But, it's half the effort of the straight forward approach. But it's also half as reliable. The conclusion from the straight forward approach is demonstrably true, the conclusion for this approach is very different. This test shows it's not demonstrably false. That conclusion has no truth value at all. It's vacuous of truth. It's an empty assertion.
Every single proof I've looked at to try to show that {} is included in { 1,2,3 } uses contraposition then assumes that the null set is a subset of everything. But it's based on assumption NOT fact. It cannot use straight forward logic. And the conclusion rendered contradicts both the defintion of a subset and the definition of null. It even contradicts the definition and stated purpose of formal logic.
And from this you accept what you called "trivial truth", but it's actually called a "vacuous truth". It's a conclusion which is accepted by formal logic as true even though it contains no truth. Let me say that again, it has NO TRUTH, but it is still accepted. The null set is accepted as a member of any set even though it has NOTHING in common with it. If this were a moral, proper, system for developing true concusions, this "vacuous" conclusion would be considered false. But you have been arguing the opposite. Denying falsehood because it cannot be proven untrue is the basis for all manner of immoral and deceptive practices. If this were accepted as proper, there would be no justice anywhere. Each person would be guilty until proven innocent.
Again, these words "vacuous truth" exist in the wikipedia article you quoted multiple times yesterday, but, you omitted it from the copy-paste. It's included within the statements you copied, one would need to intentionally omit it. Again, all of this points to a lack of moral integrity. The same sort of lack which cannot see a problem with saying "All Jews I know are ... " when they don't know any Jews at all. It's deceptive, sneaky, and wrong. And if this is what you are teaching children, then you are teaching them how to be immoral people. You are teaching them to cheat. This is the same lack of morals which would claim there are cute little pictures of a null set included as a subset, when there aren't Or that this complicated issue is explained on page 1 or 2 of a child's math book, when it isn't. A person with morals and integrity would admit that a vacuous conclusion, a statement which contains no truth should be considered false. But you are arguing against that.
