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The National Emergency Medical Services Association (NEMSA or National EMS Association) is a now defunct, registered labor union and Californianon-profit mutual benefit corporation. NEMSA's membership consists of emergency medical technicians (EMTs), and paramedics.

History[edit]

The National Emergency Medical Services Association (NEMSA) was founded in 2004 when dissatisfaction with union representation led to the creation of an all-EMS union.^[1][2] NEMSA's founding has always been a source of controversy, as founder Torren Colcord was found guilty in Alameda County (CA) Superior Court of fraud, using Service Employee International resources and company time to create and form NEMSA.^[3] 'It's a substantial judgment for substantial wrongdoing,' said Dan Martin, administrative vice president for SEIU-United Healthcare Workers West. 'They were on payroll, supposed to be looking out for members, and they did just the opposite. It's an example of what happens when people put their own personal ambitions above the interests of the union.'^[3]

SEIU was awarded a $442,000 judgement against Colcord,^[3] and later Colcord retained the guilty verdict but had the award reduced $300,000 on appeal. NEMSA members later paid for that judgement, plus attorney fees for the appeals case.^[4][5]

NEMSA supporters claimed that rather than a large union that paid no attention to EMS; they wanted a union that was for EMS, by EMS.^[6] NEMSA reported that represented 5672 members as a maximum number of members represented according to Federal LM-2 reports. The Sacramento Business Journal reported that NEMSA represented 5500 members at one point.^[7]

NEMSA promotes itself as 'a registered labor union and not-for-profit mutual benefit corporation that specializes in the labor representation of pre-hospital EMS Professionals such as EMTs, Paramedics, Dispatchers, Call Takers, Critical Care Nurses, Air Ambulance Flight Nurses and Paramedics, as well as EMS related support staff.'^[8] NEMSA represents members in 6 states.

Membership changes[edit]

Beginning in 2007, under the leadership of President Torren Colcord,^[7] NEMSA began to see dramatic changes in membership.^[7] In 2007, after difficult contract negotiations in Portland, Oregon, NEMSA disclaimed the Portland, Oregon bargaining unit, walking away from representing approximately 500 EMS workers there instead of having their recently negotiated contract rejected.^[7]

NEMSA under the leadership of Colcord lost another 1000-1500 East Coast members when those members opted to leave NEMSA in favor of the Teamsters and United EMS Workers-AFSCME Local 4911 in 2009 and 2012^[9][10]

In 2011, NEMSA under the leadership of Colcord lost 140 members in Solano County CA in favor of United EMS Workers-AFSCME Local 4911^[11]

In 2012, NEMSA under the leadership of Colcord lost 260 members in Santa Clara County CA in favor of United EMS Workers-AFSCME Local 4911^[12]

NEMSA under the leadership of Colcord lost another 2000 members in Northern California in 2012 when those members opted to leave NEMSA in favor of United EMS Workers-AFSCME Local 4911 after a coast to coast victory.^[13]

NEMSA under the leadership of Colcord lost another 140 members in Northern California at First Responder EMS Inc of Sacramento in favor of United EMS Workers-AFSCME Local 4911.

NEMSA under the leadership of Colcord lost another 75 members in Northern California at American Medical Response in favor of IAFF.

NEMSA under the leadership of Colcord lost another 350 members in New Haven CT at American Medical Response in favor of IAEP.

NEMSA lost another 500 members to NAGE in San Diego CA at Rural Metro.^[14]

NEMSA lost another 30 members from Morrow County EMS to IAEP in Mt. Gilead, Ohio.^[15]

Leadership dispute and formation of breakaway union[edit]

In 2010, a dispute over Torren Colcord's leadership of NEMSA led to a long running dispute over who controlled NEMSA. In the 2010 a union officer election was held. NEMSA refused to recognize the validity of one candidate and did not allow the candidate to run for office. That candidate, Jimmy Gambone then ran a national write in ballot campaign. According to the Department of Labor, Gambone was eligible to run for office and should have been placed on the ballot. Gambone was the winner of the election by a large majority of write in votes, Larry Lucus had the second highest and Torren Colcord had the least. According to the Department of Labor,^[16] NEMSA violated federal law in refusing to seat the winning candidate. NEMSA instead seated another candidate for President, who promptly resigned. Torren Colcord, NEMSA's outgoing President, still refusing to relinquish any power, was named Executive Director and remained in control of NEMSA. New bylaws were secretly drafted after the election to allow for the Executive Director Position to be created.^[17] The DOL investigation and resolution to the complaint took nearly two years.

During the time period the complaint was investigated by the Department of Labor, Gambone and his supporters, broke in to the NEMSA offices bringing with them mattresses, boxes of food, cameras to record live with no means to shower for the week they burglarized the office. The local police were called but the burglary was given a thumbs up from the local sheriffs department. Files stolen from the burglars included NEMSA employees private financials including bank routing numbers, SSN’s, home addresses and phone numbers. Gambone and his ilk made phone calls to the employees threatening their lives, finances and employment. The employees in turn had to contact their local police departments, lifelock and feared for their families safety.^[18][19] and left the NEMSA offices with computers, files, and information.^[20] Gambone and his supporters then seized power, naming themselves the NEMSA Board of Directors and unilaterally unseating the existing NEMSA Board of Directors.^[21] Gambone and his agents then began conducting business as NEMSA while the NEMSA Board of Directors Gambone had unseated also continued conducting business as NEMSA. According to NEMSA, Gambone supporters attempted to intercept dues checks, write checks on NEMSA's bank accounts, and do business as NEMSA during this time^[22]

Gambone and his supporters went on to continue working for a rival union, United EMS Workers or UEMSW where he was later deemed unfit and fired.

Financial condition[edit]

Federal LM-2 financial reports required for labor Unions show NEMSA's financial condition as of the end of the 2012 fiscal year for NEMSA. The 2012 report shows NEMSA claimed $12.372.00 in cash on hand and assets totaling $91,338.00 for the fiscal year ending March 31, 2013. However, NEMSA also reports total liabilities of $695,236.00 including $231,997.00 unpaid in accounts payable. This leaves NEMSA with total net assets of negative $-603,898.00.^[23]

Business practices and legal proceedings[edit]

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NEMSA hired the law firm of Goyette and Associates. In legal documents filed in Sacramento Superior Court, Paul Goyette alleges he was hired as corporate counsel to NEMSA in May 2010 at hourly rates from $225 to $250 an hour. The arrangement changed in April 2011 to a fixed monthly fee of $45,000 for legal and labor representation, plus an hourly fee of $250 for political and criminal defense work. The deal included acknowledgment that $146, 872 was due from the first agreement, but the law firm agreed not to take action to collect the money unless NEMSA altered or terminated the new agreement before March 2014. If that happened, court documents allege, the money was immediately due — plus $600,000 in liquidated damages. The money wasn’t paid and the relationship was severed by NEMSA in May 2012. In court documents filed in July, Goyette seeks to collect the more than $825,000 he says is owed.^[24]

NEMSA affiliated with NAGE (National Association of Government Employees) in 2012. The affiliation did not protect NEMSA from losing the Northern California Bargaining Unit. NAGE has since severed the affiliation and filed suit against NEMSA for breach of several contracts, claiming NEMSA did not pay NAGE more than $281,000 owed to NAGE as part of those contracts. The suit also alleges that Torren Colcord breached an employment contract with NAGE. In the suit, NAGE alleges 'Shortly after NEMSA’s affiliation with NAGE commenced, NAGE discovered that NEMSA’s financial health was not what it was represented to be during the course of NEMSA’s negotiations with NAGE, and, consequently, NEMSA did not have the financial wherewithal to meet its dues obligations to NAGE. In an effort to salvage the relationship, NAGE initially agreed to rebate to NEMSA some of the dues owed to NAGE under the Affiliation Agreement to NEMSA. In all, NAGE rebated approximately $56,000 to NEMSA. NAGE also paid NEMSA an additional $15,000 in rent pursuant to the Servicing Agreement. NAGE also performed all of the obligations and services called for under the Affiliation and Servicing Agreements prior to the termination of those agreements. Nonetheless, despite financial assistance from NAGE, and despite that NAGE has met its obligations to NEMSA under the Affiliation and Servicing Agreements, NEMSA failed to meet its financial obligations to NAGE. Of the approximately $456,000 total owed by NEMSA to NAGE pursuant to the Affiliation and Servicing Agreements, NEMSA paid NAGE only approximately $175,000 and currently owes NAGE approximately $281,000.^[25]

Politics[edit]

Deutz f3m2011 service manual. NEMSA claims it does not advocate or recommend specific political candidates or parties, nor does it participate in politics.^[26]

Critics[edit]

EMS workers in Portland, Oregon are critical of NEMSA because NEMSA disclaimed their bargaining unit when NEMSA could not negotiate an acceptable contract for their workforce. NEMSA walked away from approximately 500 members, leaving them unrepresented. 'Everybody was upset,' Charlie Savoie said. 'A faction figured maybe we'd been had, and began collecting petitions to file for new representation by the Teamsters.'^[27]

Former NEMSA member Edward Gavin, Jr. filed a lawsuit against NEMSA in 2011 claiming NEMSA had breached its duty of fair representation owed to Gavin under law. Gavin alleges 'On July 8, 2011 Mr. Gavin found out that at the Step Two grievance hearing he had briefly attended, NEMSA had withdrawn his grievance rather than allowing a decision to be made. The withdrawal of the grievance at the Step Two stage of the grievance process prevented the grievance from being taken to Step Three of the process, arbitration. As a result of the grievance being withdrawn by NEMSA, Mr. Gavin lost the opportunity to have the NEMSA stewards decide whether to take the grievance to arbitration, he lost the opportunity to appeal any decision by NEMSA not to take the grievance over his termination taken to arbitration, he lost the opportunity to have his grievance taken to arbitration, and he lost the opportunity to contest the termination of his employment. The NEMSA representatives had lied to Mr. Galvin after the Step Two hearing, when they led him to believe that the grievance hearing had been litigated to conclusion and awaited a decision. Mr. Gavin had never agreed to the withdrawal of the grievance.^[28]

EMS workers in Northern California and New England are critical of NEMSA because of a two year union leadership dispute. Leaders of this anti-NEMSA movement called themselves 'the Transition Team' and are quoted on an anti-NEMSA website as saying 'NEMSA - NEMSA is corrupt & undemocratic. It is run more like a despotic personal business' ^[29] and issued a 35-page preliminary report of the problems they found with NEMSA and NEMSA's Executive Director Torren Colcord.^[30] Later this Transition Team would abandon NEMSA and create a new union, as a local affiliate of AFSCME called United EMS Workers Local 4911^[31]

Former NEMSA employees have also been critical of NEMSA. Former NEMSA Employee Jim Misercola wrote a letter describing why he no longer works for NEMSA in which he states 'My resignation is due to the extreme philosophical and ideological differences with NEMSA. NEMSA is no longer the organization that I once tirelessly promoted. Furthermore it is my opinion that NEMSA is badly mismanaged by its current President Torren Colcord'.^[32] Former Co-Founder Tim Bonifay also stated in a letter 'I helped create NEMSA because I believed in what the organization could achieve. But the NEMSA out campaigning today is not the one that I co-founded, and I would be ashamed to be a part of that organization today.^[33]'

NEMSA members, as reported by EMS Insider have been critical of NEMSA due to NEMSA not providing Board of Directors meeting minutes and/or financial information about NEMSA upon request, in addition to getting poor representation. Many cases that should have made it to arbitration never did.^[34]

References[edit]

1. ^'Med workers pull away from top union'.
2. ^'EMS Workers Vote To Join New Union, Leave SEIU'.
3. ^ ^abc'EMS union founder liable for fraud, judge says'.
4. ^'First District Court of Appeals Overturns Award of Campaign Costs'(PDF).
5. ^'Appeals court reverses $300K judgment in union squabble'.
6. ^'EMS union founder liable for fraud, judge says'.
7. ^ ^abcd'Upstart union feels some pains as it grows'.
8. ^'About NEMSA'.
9. ^'Teamster News: AMR Contract In Negotiation'.
10. ^'Coast-to-coast victories for EMS workers, New England goes United EMS Workers-AFSCME Local 4911 with 90%'.
11. ^'Another Victory for EMS, Workers at Medic Ambulance in Solano County Vote to Join United EMS Workers-AFSCME Local 4911'.
12. ^'280 Private Sector EMS Workers in Santa Clara County Vote to Join United EMS Workers-AFSCME Local 4911'.
13. ^'Paramedics join breakaway union in effort to get contract'.
14. ^'NAGE EMS Welcomes Rural/Metro San Diego - How labor unions can help improve EMS in the 21st Century'.
15. ^'National Association of Government Employees - Service Employees International Union'. www.nage.org.
16. ^'US DOL Enforcement Actions'. US Department of Labor. July 11, 2012.
17. ^'Department of Labor decides winner of paramedic union vote'.
18. ^'Internal power struggle torments union for ambulance workers'. Sacramento Business News. 2010-11-26.
19. ^'Colcord's claim: The Transition Team and other NEMSA members 'broke' into the NEMSA office, thereby committing a crime'. www.takebackouremsunion2.org.
20. ^'Jimmy Gambone And His 'transition team' Break Into, Occupy, And Steal Confidential Files From NEMSA Office'. www.nemsausa.org.
21. ^'Jimmy Gambone and Megan McNamara declare they are the Board of Directors of NEMSA'. www.nemsausa.org.
22. ^'Gents of Gambone Scheme To Intercept Member Dues, Divert NEMSA Mail, Do Business With Employers and Vendors As NEMSA'. www.nemsausa.org.
23. ^'NEMSA 2012 LM-2'.
24. ^'Emergency workers union, attorney fight over legal fees'. Sacramento Business Journal.
25. ^'VERIFIED COMPLAINT AND DEMAND FOR JURY TRIAL'.
26. ^'NEMSA'.
27. ^Robertson, Kathy (2007-10-28). 'Upstart union experiences some pain as it grows'. Portland Business Journal.
28. ^'gov.uscourts.mad.140431' – via Internet Archive.
29. ^'Takebackouremsunion2'.
30. ^'Transition Team Preliminary Report'(PDF).
31. ^'FAQ UEMSW 042512 | American Federation Of State, County And Municipal Employees | Emergency Medical Services'. Scribd.
32. ^'Misercola NEMSA Letter'(PDF).
33. ^'Bonifay NEMSA Letter'(PDF).
34. ^'EMS Insider'(PDF).

In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox, and was described as a 'paradoxical state of affairs' by Skolem (1922: p. 295).

Skolem's paradox is that every countableaxiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the theory) that there exist sets that are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first order sentence that intuitively states 'there are uncountable sets'.

A mathematical explanation of the paradox, showing that it is not a contradiction in mathematics, was given by Skolem (1922). Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic, but the result quickly came to be accepted by the mathematical community.

The philosophical implications of Skolem's paradox have received much study. One line of inquiry questions whether it is accurate to claim that any first-order sentence actually states 'there are uncountable sets'. This line of thought can be extended to question whether any set is uncountable in an absolute sense. More recently, the paper 'Models and Reality' by Hilary Putnam, and responses to it, led to renewed interest in the philosophical aspects of Skolem's result.

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Background[edit]

One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets, such as the powerset of the natural numbers, the set of real numbers, and the Cantor set. An infinite set X is countable if there is a function that gives a one-to-one correspondence between X and the natural numbers, and is uncountable if there is no such correspondence function. When Zermelo proposed his axioms for set theory in 1908, he proved Cantor's theorem from them to demonstrate their strength.

Löwenheim (1915) and Skolem (1920, 1923) proved the Löwenheim–Skolem theorem. The downward form of this theorem shows that if a countablefirst-orderaxiomatisation is satisfied by any infinite structure, then the same axioms are satisfied by some countable structure. In particular, this implies that if the first order versions of Zermelo's axioms of set theory are satisfiable, they are satisfiable in some countable model. The same is true of any consistent first order axiomatisation of set theory.

The paradoxical result and its mathematical implications[edit]

Skolem (1922) pointed out the seeming contradiction between the Löwenheim–Skolem theorem on the one hand, which implies that there is a countable model of Zermelo's axioms, and Cantor's theorem on the other hand, which states that uncountable sets exist, and which is provable from Zermelo's axioms. 'So far as I know,' Skolem writes, 'no one has called attention to this peculiar and apparently paradoxical state of affairs. By virtue of the axioms we can prove the existence of higher cardinalities.. How can it be, then, that the entire domain B [a countable model of Zermelo's axioms] can already be enumerated by means of the finite positive integers?' (Skolem 1922, p. 295, translation by Bauer-Mengelberg)

More specifically, let B be a countable model of Zermelo's axioms. Then there is some set u in B such that B satisfies the first-order formula saying that u is uncountable. For example, u could be taken as the set of real numbers in B. Now, because B is countable, there are only countably many elements c such that c ∈ u according to B, because there are only countably many elements c in B to begin with. Thus it appears that u should be countable. This is Skolem's paradox.

Skolem went on to explain why there was no contradiction. In the context of a specific model of set theory, the term 'set' does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition of countability requires that a certain one-to-one correspondence, which is itself a set, must exist. Thus it is possible to recognise that a particular set u is countable, but not countable in a particular model of set theory, because there is no set in the model that gives a one-to-one correspondence between u and the natural numbers in that model.

From an interpretation of the model into our conventional notions of these sets, this means that although u maps to an uncountable set, there are many elements in our intuitive notion of u that don't have a corresponding element in the model. The model, however, is consistent, because the absence of these elements cannot be observed through first-order logic. With u as the reals, these missing elements would correspond to undefinable numbers.

Skolem used the term 'relative' to describe this state of affairs, where the same set is included in two models of set theory, is countable in one model, and is not countable in the other model. He described this as the 'most important' result in his paper. Contemporary set theorists describe concepts that do not depend on the choice of a transitive model as absolute. From their point of view, Skolem's paradox simply shows that countability is not an absolute property in first order logic. (Kunen 1980 p. 141; Enderton 2001 p. 152; Burgess 1977 p. 406).

Skolem described his work as a critique of (first-order) set theory, intended to illustrate its weakness as a foundational system:

'I believed that it was so clear that axiomatisation in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique.' (Ebbinghaus and van Dalen, 2000, p. 147)

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Reception by the mathematical community[edit]

A central goal of early research into set theory was to find a first-order axiomatisation for set theory which was categorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem's result showed this is not possible, creating doubts about the use of set theory as a foundation of mathematics. It took some time for the theory of first-order logic to be developed enough for mathematicians to understand the cause of Skolem's result; no resolution of the paradox was widely accepted during the 1920s. Fraenkel (1928) still described the result as an antinomy:

'Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached.' (van Dalen and Ebbinghaus, 2000, p. 147).

In 1925, von Neumann presented a novel axiomatisation of set theory, which developed into NBG set theory. Very much aware of Skolem's 1922 paper, von Neumann investigated countable models of his axioms in detail. In his concluding remarks, Von Neumann comments that there is no categorical axiomatisation of set theory, or any other theory with an infinite model. Speaking of the impact of Skolem's paradox, he wrote,

'At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known.'(Ebbinghaus and van Dalen, 2000, p. 148)

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Zermelo at first considered the Skolem paradox a hoax (van Dalen and Ebbinghaus, 2000, p. 148 ff.), and spoke against it starting in 1929. Skolem's result applies only to what is now called first-order logic, but Zermelo argued against the finitarymetamathematics that underlie first-order logic (Kanamori 2004, p. 519 ff.). Zermelo argued that his axioms should instead be studied in second-order logic, a setting in which Skolem's result does not apply. Zermelo published a second-order axiomatisation in 1930 and proved several categoricity results in that context. Zermelo's further work on the foundations of set theory after Skolem's paper led to his discovery of the cumulative hierarchy and formalisation of infinitary logic (van Dalen and Ebbinghaus, 2000, note 11).

Fraenkel et al. (1973, pp. 303–304) explain why Skolem's result was so surprising to set theorists in the 1920s. Gödel's completeness theorem and the compactness theorem were not proved until 1929. These theorems illuminated the way that first-order logic behaves and established its finitary nature, although Gödel's original proof of the completeness theorem was complicated. Leon Henkin's alternative proof of the completeness theorem, which is now a standard technique for constructing countable models of a consistent first-order theory, was not presented until 1947. Thus, in 1922, the particular properties of first-order logic that permit Skolem's paradox to go through were not yet understood. It is now known that Skolem's paradox is unique to first-order logic; if set theory is studied using higher-order logic with full semantics then it does not have any countable models, due to the semantics being used.

Current mathematical opinion[edit]

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Current mathematical logicians do not view Skolem's paradox as any sort of fatal flaw in set theory. Kleene (1967, p. 324) describes the result as 'not a paradox in the sense of outright contradiction, but rather a kind of anomaly'. After surveying Skolem's argument that the result is not contradictory, Kleene concludes 'there is no absolute notion of countability.' Hunter (1971, p. 208) describes the contradiction as 'hardly even a paradox'. Fraenkel et al. (1973, p. 304) explain that contemporary mathematicians are no more bothered by the lack of categoricity of first-order theories than they are bothered by the conclusion of Gödel's incompleteness theorem that no consistent, effective, and sufficiently strong set offirst-order axioms is complete.

Countable models of ZF have become common tools in the study of set theory. Forcing, for example, is often explained in terms of countable models. The fact that these countable models of ZF still satisfy the theorem that there are uncountable sets is not considered a pathology; van Heijenoort (1967) describes it as 'a novel and unexpected feature of formal systems.' (van Heijenoort 1967, p. 290)

References[edit]

• Barwise, Jon (1982) [1977]. 'An introduction to first-order logic'. In Barwise, Jon (ed.). Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland. ISBN978-0-444-86388-1.
• Bays, Timothy (2000). Reflections on Skolem's Paradox(PDF) (Ph.D. thesis). UCLA Philosophy Department.
• Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London-Oxford-New York: Oxford University Press. ISBN0-19-888087-1. Zbl0251.02001.
• van Dalen, Dirk; Ebbinghaus, Heinz-Dieter (Jun 2000). 'Zermelo and the Skolem Paradox'. The Bulletin of Symbolic Logic. 6 (2): 145–161. CiteSeerX10.1.1.137.3354. doi:10.2307/421203. JSTOR421203.
• Dragalin, A.G. (2001) [1994], 'Skolem's paradox', Encyclopedia of Mathematics, EMS Press
• Enderton, Herbert B. (2001). A Mathematical Introduction to Logic (2nd ed.). Elsevier. ISBN978-0-08-049646-7.
• Fraenkel, Abraham; Bar-Hillel, Yehoshua; Levy, Azriel; van Dalen, Dirk (1973). Foundations of Set Theory. North-Holland.
• Henkin, L. (1950). 'Completeness in the theory of types'. The Journal of Symbolic Logic. 15 (2): 81–91. doi:10.2307/2266967. JSTOR2266967.
• Kanamori, Akihiro (2004), 'Zermelo and set theory', The Bulletin of Symbolic Logic, 10 (4): 487–553, doi:10.2178/bsl/1102083759, ISSN1079-8986, JSTOR3216738, MR2136635
• Stephen Cole Kleene, (1952, 1971 with emendations, 1991 10th printing), Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY. ISBN0-444-10088-1. cf pages 420-432: § 75. Axiom systems, Skolem's paradox, the natural number sequence.
• Stephen Cole Kleene, (1967). Mathematical Logic.
• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam: North-Holland. ISBN978-0-444-85401-8.
• Löwenheim, Leopold (1915). 'Über Möglichkeiten im Relativkalkül'(PDF). Mathematische Annalen. 76 (4): 447–470. doi:10.1007/BF01458217. ISSN0025-5831.
• Moore, A.W. (1985). 'Set Theory, Skolem's Paradox and the Tractatus'. Analysis. 45 (1): 13–20. doi:10.2307/3327397. JSTOR3327397.
• Putnam, Hilary (Sep 1980). 'Models and Reality'(PDF). The Journal of Symbolic Logic. 45 (3): 464–482. doi:10.2307/2273415. JSTOR2273415.
• Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). New York: Springer Science+Business Media. doi:10.1007/978-1-4419-1221-3. ISBN978-1-4419-1220-6.
• Skolem, Thoralf (1923). 'Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre'. Matematikerkongressen i Helsingfors den 4-7 juli 1922; den femte Skandinaviska matematikerkongressen redogörelse. The fifth Scandinavian Mathematical Congress. Helsinki. pp. 217–232. OCLC23550016. English translation: Skolem, Thoralf (1961) [1922]. 'Some remarks on axiomatized set theory'. In van Heijenoort (ed.). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Translated by Stefan Bauer-Mengelberg. pp. 290–301.

External links[edit]

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