Instead of understanding Russell's paradox, cladistics believes that it can find it.
Instead of understanding that Russell's paradox means that classification is ultimately contradictory, cladistics claims that classification is ultimately consistent.
It means that instead of being consistent, cladistics is consistently inconsistent. It leads cladistics into a vain search for an infinite recursion, that is, the subjective side of Russell's paradox.
Cladistics is thus actually the up-side-down approach, or the belief that "if" does not exist, exemplified with the expression that "if if did not exist, so had the old woman shot the bear with a broom handle". An old cladist woman does indeed try to shoot the bear with the broom handle. She not only believes, but also claims and defines, that the broom handle is a gun, and then she shoots.
lördag 30 juni 2012
tisdag 26 juni 2012
On the black hole for classification called "cladistics"
The class clade is an infinite recursion that you enter when you conflate infinite class with finite class, ie, type with set, that is, when you don't "see" your own typification and subsequent categorization, and thereby can't distinguish them.
Conflation of these for conceptualization so fundamental concepts makes you turn Russell's paradox, ie, the fact that set theory leads to contradiction, up-side-down into the comprehension that contradiction instead is consistent, ie, forming a consistent class, that is, the class clade.
This comprehension is ambiguous between being correct in that clade is a consistent infinite class, but wrong in that clade is a consistent finite class, thus forming an infinite recursion between consistent and inconsistent, ie, a consistently inconsistent infinite loop, which you can't see because you're in it. Instead, you're convinced that there is a consistent solution at the end of this infinite recursion, which there thus isn't per definition.
Cladistics is thus a black hole in conceptualization that you enter when you conflate infinite class with finite class, ie, when you don't "see" your own typification and subsequent categorization, and thereby can't distinguish them. You simply don't know what you're doing.
The correct understanding of Russell's paradox is that classification is ultimately contradictory. This fact also means that process can't be classified consistently, ie, that there are always more than one consistent classification of a single process, because continuity is indistinguishable from class. Consistent classifications, in turn, can we only produce using an orthogonal system of classification, like the Linnean system. Instead ignoring Russell's paradox leads into an eternal orthogonal merry-go-round between inconsistent (contradictory) classifications, because it is infinitely recursive (per definition). Russell's paradox is thus a fact we need to relate to, not something we can ignore or a solvable problem. It is fundamentally due to the unavoidable fact that classification is orthogonal.
Conflation of these for conceptualization so fundamental concepts makes you turn Russell's paradox, ie, the fact that set theory leads to contradiction, up-side-down into the comprehension that contradiction instead is consistent, ie, forming a consistent class, that is, the class clade.
This comprehension is ambiguous between being correct in that clade is a consistent infinite class, but wrong in that clade is a consistent finite class, thus forming an infinite recursion between consistent and inconsistent, ie, a consistently inconsistent infinite loop, which you can't see because you're in it. Instead, you're convinced that there is a consistent solution at the end of this infinite recursion, which there thus isn't per definition.
Cladistics is thus a black hole in conceptualization that you enter when you conflate infinite class with finite class, ie, when you don't "see" your own typification and subsequent categorization, and thereby can't distinguish them. You simply don't know what you're doing.
The correct understanding of Russell's paradox is that classification is ultimately contradictory. This fact also means that process can't be classified consistently, ie, that there are always more than one consistent classification of a single process, because continuity is indistinguishable from class. Consistent classifications, in turn, can we only produce using an orthogonal system of classification, like the Linnean system. Instead ignoring Russell's paradox leads into an eternal orthogonal merry-go-round between inconsistent (contradictory) classifications, because it is infinitely recursive (per definition). Russell's paradox is thus a fact we need to relate to, not something we can ignore or a solvable problem. It is fundamentally due to the unavoidable fact that classification is orthogonal.
måndag 25 juni 2012
Cladists promise to deliver the "natural groups" tomorrow
Cladistics reminds me about when I once asked for the recipe for a good meal at a restaurant. The chef replied that you get it tomorrow. Similarly, cladists promise to deliver the "natural groups" tomorrow.
The most surprising in this issue is not cladists' belief in illusion itself, but that there indeed is someone out there willing to pay them for this illusionary delivery. It must be due to that they don't need it anyway.
The most surprising in this issue is not cladists' belief in illusion itself, but that there indeed is someone out there willing to pay them for this illusionary delivery. It must be due to that they don't need it anyway.
torsdag 21 juni 2012
The idea of a single "True Tree of Life" (ie, the foundation of cladistics) is both a definitional contradiction and a practical impossibility
The assumption that there is a single "True Tree of Life" to be found (ie, the foundation for cladistics)may appear intuitively self-evident (ie, axiomatically correct) to some of us, but it does in fact lead to Russell's paradox, meaning that it is a practical contradiction by being an infinite recursion (ie, lacking consistent solution). This fact forces a choice to discard either the assumption or the contradiction. Discarding the contradiction may then appear as the intuitively self-evident (ie, axiomatically correct) choice to some of us, but it instead meets the contradiction that nested "trees" in such hypothetical "True Tree of Life" (eg, of mitochondria or genes) may well be incongruent with the hypothetical "True Tree of Life" in that entities of nested trees (ie, holophyletic groups, or "clades") may well be incompatible with entities of the "True Tree of Life", and thus that entities (in a generic sense) may well be contradictory. This fact, however, together with the fact the assumption leads to Russell's paradox, actually mean that entities not just "may well" be contradictory, but that they are contradictory per definition, ie, lacking a consistent solution by being infinitely recursive.
The assumption that there is a single "True Tree of Life" (ie, the foundation for cladistics) may thus appear intuitively self-evident (ie, axiomatically correct) to some of us, but is actually contradictory with regard to both classes (ie, concepts) and entities. It is thus actually both a definitional contradiction and a practical impossibility.
Cladists have defended cladistics by that it is logically correct given its premises, but its premises (ie, resting on the axiom that there is a single "True Tree of Life" to be found) do thus prove to be both a definitional contradiction and a practical impossibility. Where does this fact place cladistics? Is it logically correct, but wrong? Can formulation of contradictory definitions that acknowledge a practical impossibility be logically correct? I would rather say that formulation of contradictory definitions is illogical in the first place.
The assumption that there is a single "True Tree of Life" (ie, the foundation for cladistics) may thus appear intuitively self-evident (ie, axiomatically correct) to some of us, but is actually contradictory with regard to both classes (ie, concepts) and entities. It is thus actually both a definitional contradiction and a practical impossibility.
Cladists have defended cladistics by that it is logically correct given its premises, but its premises (ie, resting on the axiom that there is a single "True Tree of Life" to be found) do thus prove to be both a definitional contradiction and a practical impossibility. Where does this fact place cladistics? Is it logically correct, but wrong? Can formulation of contradictory definitions that acknowledge a practical impossibility be logically correct? I would rather say that formulation of contradictory definitions is illogical in the first place.
tisdag 19 juni 2012
Russell's paradox and Hennig's spin of it (ie, cladistics)
Bertrand Russell demonstrated in 1901 that classification leads to contradiction (called Russell's paradox), which can be exemplified with Barber's paradox.
Barber's paradox read (from Wikipedia):
Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven by doing exactly one of two things:
All men in this town thus keep themselves clean-shaven, but whereas some shave themselves, the rest goes to the Barber, and the Barber himself consistently belongs to the other of these two groups if he behaves as defined for any of them. The Barber is thus consistently contradictory between the definitions of the two groups. This is an example of an infinite recursion. Every categorization of the Barber points to the alternative categorization in an infinite loop.
The generic reason for this kind of contradiction (ie, Russell's paradox in a generic sense) is that classification functions by distinction of difference in a similarity, since difference and similarity are diametrically opposed, ie, orthogonal, per definition, because orthogonality is contradictory per definition (in this case between the differences of the similarity). This inherent contradiction of classification is, however, invisible for us as long as it is restricted to only one of the two sides of the fundamental distinction, that is, to either process or pattern, since we then comprehend it as a similarity (rather than a difference and thus a contradiction), but becomes visible as soon as it bridges this fundamental distinction by that it then is not similar (as we intuitively expect), but instead opposite - ie, there is indeed a difference between process and pattern, but this difference is not in a similarity, but in an opposition, ie, what the Barber does is in opposition to each of the definitions of what the Barber can do.
This fundamental orthogonality of classification is also the driving force for the never-ending change in biological systematics. It is the obstacle that hinders it from reaching an unambiguous classification. However, about 50 years ago the German entomologist Willi Hennig escaped this fundamental contradiction by turning the orthogonality up-side-down. He simply asserted that only groups such as the Barber and the categories "those men in town who do not shave themselves" and "those men in town who do shave themselves" are "natural groups". He thus "acknowledged" the "difference part" (ie, contradiction) of the orthogonality and only it (ie, "denied" the "similarity part" of it). By this, he also "acknowledged" the paradoxical applications of this contradiction.
Hennig's irrational move was never accepted for publication by any scientific journal, but was instead published as a book in the 1950-ies, from which it was dragged into biological systematics in a joint effort by Steve Farris and Gareth Nelson in the 1970-ies, then won a popularity contest against consistent traditional science in the 1980-ies, and does today penetrate the thinking in biological systematics (under the name "cladistics") to the degree that the discipline is actually searching for such paradoxes (called clades) in a shared belief in the existence of a single "True Tree of Life". This imaginary "True Tree of Life" is thus actually Russell's paradox in disguise.
So, how does Hennig's move work? The answer emerges when we consider the fact that the Barber in the paradox represents the abstract similarity of a classification, whereas the two categories represent the real different parts of the classification. This consideration allows us understand that Hennig's grouping of the two categories into the Barber folds the difference itself between the real (different) parts of a classification back into its similarity, thus merely running the process of classification backwards, ie, canceling it. By this, it merely legitimates a comprehension that any classification is a "natural group". It thus does not solve the problem (ie, fact) that classification is fundamentally contradictory (and that every classification thus is contradictory per definition), but merely legitimates a comprehension that any classification is a "natural group". It does not change the fact that classification is fundamentally contradictory, but merely acknowledges contradiction, and only contradiction, as "natural groups".
The problem with this approach is that the fact that classification is orthogonal (and thus ultimately leading to Russell's paradox) means that it ultimately leads to infinite recursion of the same kind as that of Barber's paradox, today called clades, wherein every specific classification thus points to another classification in an infinite loop. (This ought not come as a surprise, since only acknowledging contradiction can't, of course, find a non-contradictory solution). The approach is thus merely a spin of classification into infinite recursion, although presently a mass-spin in biological systematics, which in practice is analogous to that like the hamster enter the running around in his wheel instead of using it as a wheel.
A consistent system of classification using an orthogonal arrangement of classes, like the Linnean system, avoids its fundamental contradiction and thus also infinite recursion, although neither it can reach unambiguity in relation to reality. Russell's paradox is namely, unfortunately, a fact we can't change.
Barber's paradox read (from Wikipedia):
Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven by doing exactly one of two things:
- Shaving himself, or
- going to the barber.
- The barber shaves only those men in town who do not shave themselves.
- Who shaves the barber?
- himself, or
- the barber (which happens to be himself).
- If the barber does shave himself, then the barber (himself) must not shave himself.
- If the barber does not shave himself, then he (the barber) must shave himself.
All men in this town thus keep themselves clean-shaven, but whereas some shave themselves, the rest goes to the Barber, and the Barber himself consistently belongs to the other of these two groups if he behaves as defined for any of them. The Barber is thus consistently contradictory between the definitions of the two groups. This is an example of an infinite recursion. Every categorization of the Barber points to the alternative categorization in an infinite loop.
The generic reason for this kind of contradiction (ie, Russell's paradox in a generic sense) is that classification functions by distinction of difference in a similarity, since difference and similarity are diametrically opposed, ie, orthogonal, per definition, because orthogonality is contradictory per definition (in this case between the differences of the similarity). This inherent contradiction of classification is, however, invisible for us as long as it is restricted to only one of the two sides of the fundamental distinction, that is, to either process or pattern, since we then comprehend it as a similarity (rather than a difference and thus a contradiction), but becomes visible as soon as it bridges this fundamental distinction by that it then is not similar (as we intuitively expect), but instead opposite - ie, there is indeed a difference between process and pattern, but this difference is not in a similarity, but in an opposition, ie, what the Barber does is in opposition to each of the definitions of what the Barber can do.
This fundamental orthogonality of classification is also the driving force for the never-ending change in biological systematics. It is the obstacle that hinders it from reaching an unambiguous classification. However, about 50 years ago the German entomologist Willi Hennig escaped this fundamental contradiction by turning the orthogonality up-side-down. He simply asserted that only groups such as the Barber and the categories "those men in town who do not shave themselves" and "those men in town who do shave themselves" are "natural groups". He thus "acknowledged" the "difference part" (ie, contradiction) of the orthogonality and only it (ie, "denied" the "similarity part" of it). By this, he also "acknowledged" the paradoxical applications of this contradiction.
Hennig's irrational move was never accepted for publication by any scientific journal, but was instead published as a book in the 1950-ies, from which it was dragged into biological systematics in a joint effort by Steve Farris and Gareth Nelson in the 1970-ies, then won a popularity contest against consistent traditional science in the 1980-ies, and does today penetrate the thinking in biological systematics (under the name "cladistics") to the degree that the discipline is actually searching for such paradoxes (called clades) in a shared belief in the existence of a single "True Tree of Life". This imaginary "True Tree of Life" is thus actually Russell's paradox in disguise.
So, how does Hennig's move work? The answer emerges when we consider the fact that the Barber in the paradox represents the abstract similarity of a classification, whereas the two categories represent the real different parts of the classification. This consideration allows us understand that Hennig's grouping of the two categories into the Barber folds the difference itself between the real (different) parts of a classification back into its similarity, thus merely running the process of classification backwards, ie, canceling it. By this, it merely legitimates a comprehension that any classification is a "natural group". It thus does not solve the problem (ie, fact) that classification is fundamentally contradictory (and that every classification thus is contradictory per definition), but merely legitimates a comprehension that any classification is a "natural group". It does not change the fact that classification is fundamentally contradictory, but merely acknowledges contradiction, and only contradiction, as "natural groups".
The problem with this approach is that the fact that classification is orthogonal (and thus ultimately leading to Russell's paradox) means that it ultimately leads to infinite recursion of the same kind as that of Barber's paradox, today called clades, wherein every specific classification thus points to another classification in an infinite loop. (This ought not come as a surprise, since only acknowledging contradiction can't, of course, find a non-contradictory solution). The approach is thus merely a spin of classification into infinite recursion, although presently a mass-spin in biological systematics, which in practice is analogous to that like the hamster enter the running around in his wheel instead of using it as a wheel.
A consistent system of classification using an orthogonal arrangement of classes, like the Linnean system, avoids its fundamental contradiction and thus also infinite recursion, although neither it can reach unambiguity in relation to reality. Russell's paradox is namely, unfortunately, a fact we can't change.
söndag 17 juni 2012
The class clade is an infinite recursion, ie, an infinite loop
The class clade is a set that includes itself as a member, meaning that it is an infinite recursion. It is actually the same process as a single clade illustrates, that is, a dichotomous propagation, run backwards. As such it is infinitely contradictory, ie, loops in an infinite contradiction.
It means that cladistics can only produce an infinite sequence of contradictory clades. Its search for clades just goes around, and around, and around....all possible contradictions.
It means that cladistics can only produce an infinite sequence of contradictory clades. Its search for clades just goes around, and around, and around....all possible contradictions.
lördag 16 juni 2012
Cladistics is in practice nothing but a way to cheat your money
Cladistics is the approach in biological systematics that only acknowledges "clades", ie, the class clade. The class clade is a set that includes itself as a member, and is thus an infinitely recursive set. An infinitely recursive set is a set that lacks a consistent solution per definition, since it is infinitely contradictory. It means that cladistics is "The Approach that Only Acknowledges Infinite Contradiction".
What this approach possibly can be good for is thus hidden in the fog of biological systematics, but cladists obtain apparently nonetheless pay for their "work". This pay can thus only be for an infinite production of contradictory clades. This money could do much more good for humanity if they were invested in something else than cladistics. Cladistics is in practice nothing but a way to cheat your money.
What this approach possibly can be good for is thus hidden in the fog of biological systematics, but cladists obtain apparently nonetheless pay for their "work". This pay can thus only be for an infinite production of contradictory clades. This money could do much more good for humanity if they were invested in something else than cladistics. Cladistics is in practice nothing but a way to cheat your money.
fredag 15 juni 2012
On the contradiction in the idea of a single "True Tree of Life"
The (today cladistic) idea of a single "True Tree of Life" is contradictory between being a single entity or many entities, ie, between being one or many (independently of what).
This contradiction is, however, ambiguous in that it can be consistently folded backwards until ending in a single entitity, thus making the idea appear as a "natural" (and thereby axiomatically consistent) idea of the origin of many entities from a single entity (thus at the same time as it is contradictory between one or many).
Now, if this "folding backwards" indeed is consistent, and the idea thus is a consistent idea of the origin of many entities from a single entity, then the idea is obviously contradictory between being consistent or inconsistent.
Understanding of this contradiction (actually Russell's paradox) resides in that the idea is contradictory in terms of being one or many entities of the same kind, but consistent in terms of being of the same kind, because a class (ie, kind) is both infinite and finite per definition, and is thus contradictory between being inconsistent or consistent per definition. The idea is thus contradictory between being consistent or inconsistent simply because classes are contradictory between being inconsistent or consistent. The idea is thus a practical impossibility, ie, an illusion, that emerges when we believe that kinds (ie, classes) are real (ie, form consistent entities), because such "entities" are contradictory between being one or many, and are thus incompatible with (actually orthogonal to) real entities (eg, organisms).
The (today cladistic) idea of a single "True Tree of Life" is thus a practical impossibility, ie, an illusion, that emerges when we believe that kinds (ie, classes) are real (ie, form consistent entities), because classes are fundamentally contradictory between one and many (as Bertrand Russell also showed in 1901 with Russell's paradox).
Had we all been aware of human progress, cladistics should thus not have emerged.
This contradiction is, however, ambiguous in that it can be consistently folded backwards until ending in a single entitity, thus making the idea appear as a "natural" (and thereby axiomatically consistent) idea of the origin of many entities from a single entity (thus at the same time as it is contradictory between one or many).
Now, if this "folding backwards" indeed is consistent, and the idea thus is a consistent idea of the origin of many entities from a single entity, then the idea is obviously contradictory between being consistent or inconsistent.
Understanding of this contradiction (actually Russell's paradox) resides in that the idea is contradictory in terms of being one or many entities of the same kind, but consistent in terms of being of the same kind, because a class (ie, kind) is both infinite and finite per definition, and is thus contradictory between being inconsistent or consistent per definition. The idea is thus contradictory between being consistent or inconsistent simply because classes are contradictory between being inconsistent or consistent. The idea is thus a practical impossibility, ie, an illusion, that emerges when we believe that kinds (ie, classes) are real (ie, form consistent entities), because such "entities" are contradictory between being one or many, and are thus incompatible with (actually orthogonal to) real entities (eg, organisms).
The (today cladistic) idea of a single "True Tree of Life" is thus a practical impossibility, ie, an illusion, that emerges when we believe that kinds (ie, classes) are real (ie, form consistent entities), because classes are fundamentally contradictory between one and many (as Bertrand Russell also showed in 1901 with Russell's paradox).
Had we all been aware of human progress, cladistics should thus not have emerged.
måndag 11 juni 2012
Om Livets Träd (och kladistik)
Sedan slutet av 1960-talet har den biologiska systematiken alltmer börjat överge den Linneanska systematiken för att istället tala om "naturliga grupper" och det man kallar "Livets träd" företrätt av en åsiktsinriktning som kallas kladistik. Denna idé är inte ny, utan fanns bevisligen redan hos Taoismens grundare, de antika grekerna, vikingarna och en massa andra kulturer och har levt parallella liv hela tiden sedan dess, för att nu återigen dyka upp i den moderna biologiska systematiken.
Exempel är:
För massor av ytterligare illustrationer, se Livets träd).
Idag kallas sådana släktträd dock "kladogram", varav det allomfattande kladogrammet alltså är livets träd.
Ett exempel är:
Problemet med denna idé är dock att den är inkonsekvent, dvs självmotsägande, som en singularitet (dvs ett enda Livets Träd) genom att innebära att enheterna det består av måste vara både åtskilda och överlappande samtidigt. Idén är helt enkelt i praktiken en direkt sammanblandning av vår särskiljning av enheterna det består av, oavsett vilka enheter vi än särskiljer för att konstruera det. Den är alltså i praktiken en självmotsägande cirkularitet, trots att många av oss tvärtom uppfattar den som "naturlig". Denna egenskap innebär att Livets Träd kan vara precis vad som helst, men alltid innehållande motsägelser. Idén är helt enkelt konsekvent inkonsekvent. (Egentligen representerar idén den subjektiva aspekten av Russell's paradox, men tar ett tag att reda ut).
Det enda sätt vi har att undvika denna paranoida cirkularitet är att dela upp enheterna trädet består av i kategorier av klasser, liksom i den Linneanska systematiken, genom att på så sätt konsekvent hålla isär begreppen klass och enhet (dvs oändlig och ändlig), då kategorierna representerar (ändliga) enheter och klasserna är oändliga.
Detta innebär att det inte finns något enda sant Livets Träd, utan endast flera lika sanna beskrivningar av Livets uppkomst. Livets uppkomst går helt enkelt att betrakta ur olika aspekter som är lika sanna. Idén om ett enda sant Livets Träd är i en vetenskaplig mening mer av en psykisk sjukdom än en realistisk idé, även om den för konsten erbjuder ett intressant uppslag (se ovan och på Livets träd).
Exempel är:
| All she requested was that the painting should depict a Tree of Life. |
| Amnioternas "fylogeni". |
För massor av ytterligare illustrationer, se Livets träd).
Idag kallas sådana släktträd dock "kladogram", varav det allomfattande kladogrammet alltså är livets träd.
Ett exempel är:
 |
| ITOL Tree of life (Wikipedia) |
Problemet med denna idé är dock att den är inkonsekvent, dvs självmotsägande, som en singularitet (dvs ett enda Livets Träd) genom att innebära att enheterna det består av måste vara både åtskilda och överlappande samtidigt. Idén är helt enkelt i praktiken en direkt sammanblandning av vår särskiljning av enheterna det består av, oavsett vilka enheter vi än särskiljer för att konstruera det. Den är alltså i praktiken en självmotsägande cirkularitet, trots att många av oss tvärtom uppfattar den som "naturlig". Denna egenskap innebär att Livets Träd kan vara precis vad som helst, men alltid innehållande motsägelser. Idén är helt enkelt konsekvent inkonsekvent. (Egentligen representerar idén den subjektiva aspekten av Russell's paradox, men tar ett tag att reda ut).
Det enda sätt vi har att undvika denna paranoida cirkularitet är att dela upp enheterna trädet består av i kategorier av klasser, liksom i den Linneanska systematiken, genom att på så sätt konsekvent hålla isär begreppen klass och enhet (dvs oändlig och ändlig), då kategorierna representerar (ändliga) enheter och klasserna är oändliga.
Detta innebär att det inte finns något enda sant Livets Träd, utan endast flera lika sanna beskrivningar av Livets uppkomst. Livets uppkomst går helt enkelt att betrakta ur olika aspekter som är lika sanna. Idén om ett enda sant Livets Träd är i en vetenskaplig mening mer av en psykisk sjukdom än en realistisk idé, även om den för konsten erbjuder ett intressant uppslag (se ovan och på Livets träd).
fredag 8 juni 2012
How do we cure a fanatic, like a cladist?
The class clade does indeed appear as the only "natural groups" to those of us that have a strong tendency for typification (like the German entomologist Willi Hennig), but the problem is that the class is contradictory. Since there are clades of different kinds of things, which also may be physical parts of each other, single clades are indeterminate as to whether being one or many both in and over time. This contradiction was described by Bertrand Russell in 1901 and is therefore called Russell's paradox.
Today's cladists thus neither understand that the class clade is contradictory nor are aware of Russell's paradox, but instead believe, actually assert, claim and define that the class can be found. They appear caught in the paradox.
Who on earth can make such convinced believers (ie, fundamentalists) realize that they are simply wrong? Or as Amos Oz put it: how do we cure a fanatic?
Today's cladists thus neither understand that the class clade is contradictory nor are aware of Russell's paradox, but instead believe, actually assert, claim and define that the class can be found. They appear caught in the paradox.
Who on earth can make such convinced believers (ie, fundamentalists) realize that they are simply wrong? Or as Amos Oz put it: how do we cure a fanatic?
tisdag 5 juni 2012
Is not cladistics rather paradoxical?
The British philosopher, logician, mathematician and historian Bertrand Russell showed 1901 that naive set theory leads to paradox.
The German entomologist Willi Hennig asserted (claimed) 1955 that only clades (ie, historical sets) are "natural" groups and that there is a single clade of clades (ie, historical set of all historical sets, or a true tree of life) to be found.
Since Russell's paradox no doubt is true, Hennig must thus assert (claim) that paradoxes can be found.
"Finding" a paradox can only mean finding a solution of the paradox. Russell's paradox is stated as:
The paradox thus means that R (ie, the set of all sets that are not members of themselves), corresponding to Hennig's "historical set of all historical sets (that are not members of themselves)" is inconsistent.
So, did Hennig find a consistent solution of this paradox? No, he merely asserted (claimed) that only such paradoxes (ie, sets) are "natural" groups and that there is a single solution to them.
Hennig's assertion (claim) gave rise to the new branch of biological systematics called cladistics and looking for the solution to Russell's paradox.
This leads unsought to the question: is not this new branch (ie, cladistics) rather paradoxical?
The German entomologist Willi Hennig asserted (claimed) 1955 that only clades (ie, historical sets) are "natural" groups and that there is a single clade of clades (ie, historical set of all historical sets, or a true tree of life) to be found.
Since Russell's paradox no doubt is true, Hennig must thus assert (claim) that paradoxes can be found.
"Finding" a paradox can only mean finding a solution of the paradox. Russell's paradox is stated as:
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if it is not a member of itself, it qualifies as a member of itself by the same definition (slightly modified from Wikipedia).
The paradox thus means that R (ie, the set of all sets that are not members of themselves), corresponding to Hennig's "historical set of all historical sets (that are not members of themselves)" is inconsistent.
So, did Hennig find a consistent solution of this paradox? No, he merely asserted (claimed) that only such paradoxes (ie, sets) are "natural" groups and that there is a single solution to them.
Hennig's assertion (claim) gave rise to the new branch of biological systematics called cladistics and looking for the solution to Russell's paradox.
This leads unsought to the question: is not this new branch (ie, cladistics) rather paradoxical?
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