I promised an explanation of Russell's paradox and how I believe it can be used outside of a purely mathematical context.

Interested parties should of course read the explanation on wiki or your favorite reference, but I'll give my informal (simplified) synopsis with what I see as the key features.

Naive set theory, as proposed in informal language, asserts Cantor's definition of set as "a gathering together into a whole of definite, distinct objects of our perception or of our thought." Although the logical language of set theory is first-order predicate logic, at this level of description we can use the simpler form of propositional logic: a set is well defined if we can formulate a propositional formula that describes its elements. That is, because the logical definition of a propositional formula P(x) is that for every x we can say (in principle) that P(x) is true or false but not both.

It follows from the assertion of this "definition" of set that the propositional formula* P(x) iff x is a set* is a well defined proposition - otherwise the notion of set would be poorly defined. But because *P(x) iff x is a set* is assumed to be a well formed proposition, by the definition of set it must define a set, i.e., the set of all things that are sets. At this point I'll note that we aren't really using many of the properties of sets. What we've really done is piggyback on the logical foundation to assert the existence of some collections of things that correspond with the action of propositions on them.

However, now that we have the set of all sets, we do need to assert properties of membership. Still we do not need the full machinery of Cantor's set theory to generate a paradox, only some rather intuitive rules about things that are members of collections. Specifically, if a collection of things or a set is well defined, and I take some of the members of that collection, the collection I have taken inherits the well-definition of the original collection. In talking about sets, a subset of a set is also a set.

And at this point we can also assert that the proposition of self-membership must be well defined, since the members of a set are definite. So whether a set has a particular member must be propositionally true or false, and since the set of all sets is itself a set, it follows that it is a member of itself.

Given rules of propositional logic, the negation of a proposition is also a proposition, so we can form a proposition*S(x) iff x is not a member of itself*, which derives from the negation of the proposition *P(x) iff x is a member of itself* - a proposition already established as well defined. Now we get another subset of the set of all sets, the set of all sets that are not members of themselves. This must be a set since it is a subset of the collection of all sets.

Note that the key move here is this assertion that the aggregate of all objects of a given type is itself an object of the same type.

At this point, the collection of all sets that are not members of themselves - In mathematical language,*S = { X | X is not an element of X }* - generates a paradox when we ask, is S a member of itself?

1. Start by assuming S is not a member of itself. In that case, it satisfies the definition of membership in S which is defined as the set of all sets that do not belong to themselves. That means S must be a member of S, i.e. it does belong to itself, which contradicts the assumption we started out with.

2. Suppose S is a member of itself. Then it must satisfy the conditions for membership in S which is defined as the set of all sets that do not belong to themselves. Thus we conclude that S must not be a member of itself, which again contradicts the assumption we started out with.

______

Although much deeper predicate logic can be used to describe this, for the purposes of general discussion, this paradox does not really require much beyond:

1. Naive use of propositional logic in a free universe of consideration (notice we never scoped our universe of consideration to objects of a particular type - we were free to consider anything that might be encoded in a propositional formula).

2. A rudimentary notion of membership directly tied to propositional formulas.

Even the notion of self-membership (which is where things seem to start going off the rails) is necessitated by (1) and (2), which give us the assumption that the "collection of all collections" or set of all sets is well defined.

_______

So when I encounter mystical notions such as a God that is the "transcendent summation of all that exists", I see a very similar construction to naive set theory:

1. Is*P(x) iff x exists* a valid propositional formula? If not, we can stop right here and say our logical foundations are simply unusable. In this case, the statement that the God under discussion "exists" or perhaps does not is itself incoherent.

2. On the other hand, if we do (as most would) assume that existence is a well defined propositional property, then this God that contains the transcendent culmination of all that exists has properties very much like the set of all sets, and allows us to construct an analog of Russell's paradox.

To me, it's also a giveaway that type has become mixed here. If we postulate a thing that subsumes the entirety of existence, it seems suspiciously odd that this monstrous aggregation would have any properties in common with its lowly members. This is where the transcendent God appears to contradict a loving personal God. Suppose we define some other property such as*P(x) iff x is capable of feeling love*. Is the set or collection of all x that are capable of feeling love itself capable of feeling love? If we think it might, we're back in a set of all sets situation, and can very likely construct a paradox.

I suspect this prohibition on free mixing of type is a general principle. Can the free aggregation of all objects with a particular property itself be an object with that property? If so, it seems we'd be able to construct another Russell-type paradox.

What do you think? Is this paradox of use in general discourse? Or does it belong solely in Set Theory class?

]]>Interested parties should of course read the explanation on wiki or your favorite reference, but I'll give my informal (simplified) synopsis with what I see as the key features.

Naive set theory, as proposed in informal language, asserts Cantor's definition of set as "a gathering together into a whole of definite, distinct objects of our perception or of our thought." Although the logical language of set theory is first-order predicate logic, at this level of description we can use the simpler form of propositional logic: a set is well defined if we can formulate a propositional formula that describes its elements. That is, because the logical definition of a propositional formula P(x) is that for every x we can say (in principle) that P(x) is true or false but not both.

It follows from the assertion of this "definition" of set that the propositional formula

However, now that we have the set of all sets, we do need to assert properties of membership. Still we do not need the full machinery of Cantor's set theory to generate a paradox, only some rather intuitive rules about things that are members of collections. Specifically, if a collection of things or a set is well defined, and I take some of the members of that collection, the collection I have taken inherits the well-definition of the original collection. In talking about sets, a subset of a set is also a set.

And at this point we can also assert that the proposition of self-membership must be well defined, since the members of a set are definite. So whether a set has a particular member must be propositionally true or false, and since the set of all sets is itself a set, it follows that it is a member of itself.

Given rules of propositional logic, the negation of a proposition is also a proposition, so we can form a proposition

Note that the key move here is this assertion that the aggregate of all objects of a given type is itself an object of the same type.

At this point, the collection of all sets that are not members of themselves - In mathematical language,

1. Start by assuming S is not a member of itself. In that case, it satisfies the definition of membership in S which is defined as the set of all sets that do not belong to themselves. That means S must be a member of S, i.e. it does belong to itself, which contradicts the assumption we started out with.

2. Suppose S is a member of itself. Then it must satisfy the conditions for membership in S which is defined as the set of all sets that do not belong to themselves. Thus we conclude that S must not be a member of itself, which again contradicts the assumption we started out with.

______

Although much deeper predicate logic can be used to describe this, for the purposes of general discussion, this paradox does not really require much beyond:

1. Naive use of propositional logic in a free universe of consideration (notice we never scoped our universe of consideration to objects of a particular type - we were free to consider anything that might be encoded in a propositional formula).

2. A rudimentary notion of membership directly tied to propositional formulas.

Even the notion of self-membership (which is where things seem to start going off the rails) is necessitated by (1) and (2), which give us the assumption that the "collection of all collections" or set of all sets is well defined.

_______

So when I encounter mystical notions such as a God that is the "transcendent summation of all that exists", I see a very similar construction to naive set theory:

1. Is

2. On the other hand, if we do (as most would) assume that existence is a well defined propositional property, then this God that contains the transcendent culmination of all that exists has properties very much like the set of all sets, and allows us to construct an analog of Russell's paradox.

To me, it's also a giveaway that type has become mixed here. If we postulate a thing that subsumes the entirety of existence, it seems suspiciously odd that this monstrous aggregation would have any properties in common with its lowly members. This is where the transcendent God appears to contradict a loving personal God. Suppose we define some other property such as

I suspect this prohibition on free mixing of type is a general principle. Can the free aggregation of all objects with a particular property itself be an object with that property? If so, it seems we'd be able to construct another Russell-type paradox.

What do you think? Is this paradox of use in general discourse? Or does it belong solely in Set Theory class?

Which commandments do you think are superior; Yahweh’s or Gnostic Christianity’s?

You may use whatever set of commandments you think Yahweh gave. There are a number of renditions.

As for the Gnostic commandments, I offer the following.

1. You shall place no commandments above theseunless proven to be morally superior.

2. You shall value all people as equal before thelaw. The inequality of outcome is punishment enough of itself.

3. You shall live by the golden rule and respondwith reciprocity of harm or care to what is done to you.

4. Use Gnosis and put logic and reason and theirproofs above faith, which by its nature has no proofs, logic or reason.

5. You shall leave the environment in a bettercondition than what is given to you as an inheritance to your next generation.

6. You shall not impoverish the next generation andlive according to the means you produce as their labor and wealth is theirs andnot yours to squander.

Gnostic Christianity and free thinking lost the God wars when the Orthodox Church decimated us and burned most of our scriptures. I think that Gnostic Christians had a superior set of commandments then as well as now. Those commandments were not only meant for seekers after a God but also a guide to secular law. Both secular law and Christianity seemed to ignore the second commandment of equality till our modern era. As a Gnostic Christian, I ask (rhetorically), what took the world so long to catch up to Gnostic Christian thinking and what is Islam and other backwards thinking people waiting for.

Many have a problem with the 10 commandments given by Yahweh so I thought I would see if there is a consensus of thought on the Gnostic Christian ideology as compared to the Christian ideology. The main complaints I see are that Yahweh’s commandments have created a Christian ideology that denies gays and women equality. I think all souls to be created equal and thus deserving of equal human statue and citizenship.

Others as seen in these two link have their own views and I would add that I think Yahweh’s no divorce policy, --- which Jesus confirms.--- and Yahweh’s policy of accepting bribes, ransoms or sacrifices (these a reall analogue) to alter his usual and moral policy punishing the guilty and not the innocent, --- to the immoral policy of punishing the innocent instead ofthe guilty, as exemplified by his accepting Jesus as a sacrifice to save sinners whom God himself created to be sinners.

https://www.youtube.com/watch?v=8u3z69YpLx0#t=100

https://www.youtube.com/watch?v=BUfGRN4HVrQ

Thanks in advance for your reply.

Regards

DL

]]>You may use whatever set of commandments you think Yahweh gave. There are a number of renditions.

As for the Gnostic commandments, I offer the following.

1. You shall place no commandments above theseunless proven to be morally superior.

2. You shall value all people as equal before thelaw. The inequality of outcome is punishment enough of itself.

3. You shall live by the golden rule and respondwith reciprocity of harm or care to what is done to you.

4. Use Gnosis and put logic and reason and theirproofs above faith, which by its nature has no proofs, logic or reason.

5. You shall leave the environment in a bettercondition than what is given to you as an inheritance to your next generation.

6. You shall not impoverish the next generation andlive according to the means you produce as their labor and wealth is theirs andnot yours to squander.

Gnostic Christianity and free thinking lost the God wars when the Orthodox Church decimated us and burned most of our scriptures. I think that Gnostic Christians had a superior set of commandments then as well as now. Those commandments were not only meant for seekers after a God but also a guide to secular law. Both secular law and Christianity seemed to ignore the second commandment of equality till our modern era. As a Gnostic Christian, I ask (rhetorically), what took the world so long to catch up to Gnostic Christian thinking and what is Islam and other backwards thinking people waiting for.

Many have a problem with the 10 commandments given by Yahweh so I thought I would see if there is a consensus of thought on the Gnostic Christian ideology as compared to the Christian ideology. The main complaints I see are that Yahweh’s commandments have created a Christian ideology that denies gays and women equality. I think all souls to be created equal and thus deserving of equal human statue and citizenship.

Others as seen in these two link have their own views and I would add that I think Yahweh’s no divorce policy, --- which Jesus confirms.--- and Yahweh’s policy of accepting bribes, ransoms or sacrifices (these a reall analogue) to alter his usual and moral policy punishing the guilty and not the innocent, --- to the immoral policy of punishing the innocent instead ofthe guilty, as exemplified by his accepting Jesus as a sacrifice to save sinners whom God himself created to be sinners.

https://www.youtube.com/watch?v=8u3z69YpLx0#t=100

https://www.youtube.com/watch?v=BUfGRN4HVrQ

Thanks in advance for your reply.

Regards

DL

Hello,

we'd like to hear your thoughts about a possible change to the D&D layout. Please find the consultation thread in the Questions & Suggestions section of the site and comment freely.

Thanks :)

]]>we'd like to hear your thoughts about a possible change to the D&D layout. Please find the consultation thread in the Questions & Suggestions section of the site and comment freely.

Thanks :)

Quote:

The Florida Attorney General's price-gouging hotline has received 4,000 calls since it opened Monday, attorney general Pam Bondi said.

The reports include essential products such as water, food, and baby items. In Tampa, there were reports of 7-Eleven stories charging $32 for cases of bottled water. And, some third-party companies were selling water through Amazon and tacking on $100 shipping fees, she said. Amazon pulled those vendors.

Though these are not criminal offenses, Bondi said her department will do what it can to discourage it, including calling out the companies publicly.

Quote:

Government Barriers to Private Solutions

"Price gouging—like spinach—may be unappealing at first bite but it's good for everyone in the long run."

I am sure this has been discussed before, but the hurricane season seems to bring out the economic idiots. Let's have another go at whether it is good or bad for private people and institutions to charge the going rate to provide goods and services that are needed and wanted. Or we can keep prices and supply lower and simply wait for the government to provide the goods and services. I am thinking more about the general rebuilding post disaster problems rather than the silly example of $$ for water the very moment water is scarce without waiting a bit for a rebalance in supply and demand. For example, is it a good or a bad thing that roofers charge quite a bit more than they would charge before the hurricane? Higher prices bring in more people to supply a needed service. Does 'Joe' want to relocate from a comfortable home in Iowa to provide roofing services for a bit in Houston without a good premium for his services?

If the wages of sin is death, and Jesus died, does that makeJesus a sinner?

To sin is to do something immoral and thus create a victimto that sin. I cannot see all sins as something that would condemn us toeternal suffering and death in hell or the lake of fire as that goes againstthe biblical notion of justice being an eye for an eye or that the penaltyshould fit the crime/sin.

What sin did Jesus do to earn his death?

I see Jesus’ death as more of suicide than sacrifice as heinitiated his own suicide by getting Judas to betray him.

Could Jesus’ sin have been suicide?

If not, what do you think his sin was?

Regards

DL

]]>To sin is to do something immoral and thus create a victimto that sin. I cannot see all sins as something that would condemn us toeternal suffering and death in hell or the lake of fire as that goes againstthe biblical notion of justice being an eye for an eye or that the penaltyshould fit the crime/sin.

What sin did Jesus do to earn his death?

I see Jesus’ death as more of suicide than sacrifice as heinitiated his own suicide by getting Judas to betray him.

Could Jesus’ sin have been suicide?

If not, what do you think his sin was?

Regards

DL