Some Christians believe in God and Jesus, but not the devil.
If so, what is their explanation of the temptation of Christ in the desert?
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For people who believe in Jesus and not the devil, what happend in the desert?
There are Christians who believe in Jesus, but not in a personal Devil. For example: the Christadelphians.
Christadelphians believe that the Satan or Devil is not an independent spiritual being or fallen angel. Devil is viewed as the general principle of evil and inclination to sin which resides in humankind. They are convinced that, dependent on the context, the term Satan in Hebrew merely means "opponent" or "adversary" and is frequently applied to human beings. Accordingly, they do not define Hell as a place of eternal torment for sinners, but as a State of Eternal Death respectively non-existence due to annihilation of body and mind. - Christadelphian (Wikipedia)
They tend not to take the Scriptures “literally”. This includes the temptations of Christ in the desert.
The temptations evidently cannot be taken literally:
Matthew 4:8 implies that Jesus was led up into a high mountain to see all the kingdoms of the world in their future glory "in a moment of time" (Mk.4:5). There is no mountain high enough to see all the world. And why would the height of the mountain enable Jesus to see what the world would be like in the future? The earth being a sphere, there is no point on its surface from which one can see all the parts of the world at any one time.
A comparison of Matt. 4 and Luke 4 shows that the temptations are described in a different order. Mark 1:13 says that Jesus was "in the wilderness forty days, tempted of Satan", whilst Matt. 4:2-3 says that "when he had fasted forty days...the tempter (Satan) came to Him...". Because Scripture cannot contradict itself, we can conclude that these same temptations kept repeating themselves. The temptation to turn stones into bread is an obvious example. This would fit nicely if these temptations occurred within the mind of Jesus. Being of our nature, the lack of food would have affected him mentally as well as physically, and thus his mind would have easily begun to imagine things. Just going a few days without food can lead to delirium for some (cp. 1 Sam.30:12). The similarity between rolls of bread and stones is mentioned by Jesus in Mt.7:9, and doubtless those images often merged in his tortured mind - although always to be brought under swift control by his recollection of the Word.
Jesus probably told the Gospel writers the record of his temptations, and to bring home in words the intensity of what he underwent, he could have used the figurative approach seen in Matt. 4 and Luke 4.
It seems unlikely that the devil led Jesus through the wilderness and streets of Jerusalem and then they scaled a pinnacle of the temple together, all in view of the inquisitive Jews. Josephus makes no record of anything like this happening - presumably it would have caused a major stir. Similarly, if these temptations occurred several times within the forty days as well as at the end of that period (which they did at least twice, seeing that Matthew and Luke have them in different order), how would Jesus have had time to walk (n.b. the devil "led" Jesus there) to the nearest high mountain (which would have been Hermon in the far north of Israel), climb to the top and back down again, return to the wilderness and then repeat the exercise? His temptations all occurred in the wilderness - he was there for forty days, tempted all the time by the devil (who only departed at the end - Matt. 4:11). If Jesus was tempted by the devil each day, and the temptations occurred only in the wilderness, then it follows that Jesus could not have left the wilderness to go to Jerusalem or travel to a high mountain. These things therefore could not have happened literally.
If the devil is a physical person who has no respect for God's Word and is interested in making people sin, then why would Jesus quote Scripture to him to overcome him? According to the popular view, this would not send the devil away. Notice that Jesus quoted a Bible passage each time. If the devil was the evil desires within Jesus' heart, then it is understandable that by his having the Word in his heart and reminding himself of it, he could overcome those bad desires. Psalm 119:11 is so relevant that perhaps it is specifically prophesying Christ's experiences in the wilderness: "Thy word have I hid in mine heart, that I might not sin against Thee".
Matt. 4:1 says that Jesus was "led up of the spirit into the wilderness to be tempted of the devil." This was the Spirit of God which had just been bestowed upon him (ch. 3:16). It would be an extraordinary thing for the Spirit of God to lead Jesus into the wilderness so that he cftould be tempted by a superhuman being existing in opposition to God. - Digression 20: The Temptations Of Jesus
Many Christians who deny the Devil will follow similar lines: Scriptures are not to be always taken literally.