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On the difference between naive set theory (like cladistics) and axiomatic set theory (like ZFC and Linnean systematics)

Tuesday, December 10, 2013 2:13
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The naive definition of a “set” is “a collection of distinct elements”. However, according to this definition, a set of the type “clade” (defined as ”an ancestral element and all of its descendant elements”)  is paradoxically contradictory between a set and an element with respect to properties (see Russell’s paradox). This definition of “set” is thus paradoxically contradictory.

Mathematics avoids this paradox by comprehending a set as an UNDEFINED PRIMITIVE, whose properties instead are defined by the Zermelo–Fraenkel axioms (ZFC). The most basic properties of a set in this context are that it ”has” elements, and that two sets are equal (one and the same) if and only if every element of one is an element of the other (compare to how also a Linnean “genus” has “species”). This trick avoids Russell’s paradox by avoiding to define “set”, thereby also avoiding to assign single sets as single elements (ie, conflating sets with elements), but instead circularly defining “element” as “an element of the set sets“. In this context a single set is not “one and the same” as itself, but only two sets can be “one and the same”, ie, if and only if every object of one is an element of the other. A single set is instead axiomatically a subset to itself (and mathematically the product of the empty set). This solution is consistent.

The “father of cladistics”, Willi Hennig, however, appears to have been totally unaware of this set theory problem, instead making one of the most common logical errors, that is, to assume that the converse (or the inverse) of a conditional statement is equivalent to this conditional statement, thereby simply conflating “set” with “element” as in the naive definition of “set” above. More clearly spelled out, his logical error is to assume that the implication “if p then q” also implies “if q then p“, ie, that the implication is an equality between p and q, in practice, that the implication from “element” to “set” is an equality between them.

However, Hennig’s logical error was so popular among biological systematists that it took over biological systematics under the name of “Cladistics”. It means that most biological systematists today are explicitly illogical. They do not avoid Russell’s paradox (as ZFC and and the Linnean systematics do), but instead search it using very fancy algorithms, as if a paradox can be found, when it in practice is an infinite recursion, ie, infinite loop. The question is thus: what can stop biological systematists (ie, cladists) from this vain search? Any suggestions…?



Source: http://menvall.wordpress.com/2013/12/10/on-the-difference-between-naive-set-theory-like-cladistics-and-axiomatic-set-theory-like-zfc-and-linnean-systematics/

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